Modelling and Control of Reactive Distillation Processes
Nicholas C. T. Biller
A thesis submitted for the degree of Doctor of Philosophy of the University of London
Department of Chemical Engineering University College London London WCIE 7JE
September 2003 ProQuest Number: 10014376
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Reactive distillation has been applied successfully in industry where large capital and energy savings have been made through the integration of reaction and distillation into one system. Operating in batch mode, in either tray or packed columns, offers the flexibihty required by pharmaceutical and hne chemical industries for producing low volume/high value products with varying specifications. However, regular packed or tray columns may not be suitable for high vacuum operations due to the pressure drop across the column section and short path distillation may be more applicable.
The objective of this thesis is to investigate the control of reactive distillation in batch columns, tray and packed, and in short path columns. In order to study control fully, it is necessary to develop rigorous dynamic models that accurately capture the process behaviour. The higher the degree of rigour, the more accurately the process conditions and dynamics are captured. However, more rigorous models are more computationally expensive to implement and can be prone to numerical errors, introduced for instance during linearisation. Therefore, in this thesis, the degree of modelling rigour required for both simulation and control purposes is explored in detail for tray and packed batch columns and short-path columns.
For batch tray columns, it is demonstrated that to accurately capture the change in process conditions, it is necessary to model pressure dynamics and employ a dynamic energy balance. For packed columns, distributed rate based modelling is compared to lumped equilibrium modelling and it is found that due to the varying conditions within the packing, the efficiency changes, resulting in mismatch between the two methods. The short-path distillation column which has hitherto only been modelled at steady-state, is modelled using a dynamic rate based model, essential for investigating control.
Having developed the dynamic models, the control and controllability of these reactive distillation processes are examined. General control properties of reactive batch distillation are discussed and methods are presented for applying linear controllability tools to these non-linear process models. The linear models are then employed to demonstrate the implications for control when adopting one of the three processes. Acknowledgements
I would like to thank my supervisor, Dr. Eva Sprensen, for her guidance and encouragement througout the course of this work. I would also like to thank the students and staff, past and present, of the Computer Aided Process Engineering group for the lively and useful discussions and the occasional, but necessary, excursions to Huntley St. The financial support from the Engineering and Physical Sciences Research Council and the Centre for
Process Systems Engineering is gratefully acknowledged. Finally, I would like to thank
Anna for her constant love and support. C ontents
Abstract 2
Acknowledgements 3
List of figures 10
List of tables 14
1 Introduction 15
1.1 M otivation ...... 15
1.2 Reactive distillation ...... 16
1.3 Batch column operation ...... 17
1.3.1 Control of reactive batch distillation ...... 18
1.4 Short-path column operation ...... 19
1.4.1 Falling film e v a p o ra to rs...... 19
1.4.2 Wiped him evaporators ...... 20
1.4.3 Short-path distillation ...... 20
1.5 Objectives of this w o rk...... 21
1.6 Outline of this t h e s i s ...... 22
1.7 Main contributions ...... 23
4 CONTENTS 5
2 Literature review 25
2.1 Modelling of reactive batch distillation ...... 25
2.1.1 Introduction ...... 25
2.1.2 Reactive batch distillation literature ...... 26
2.1.3 Reactive batch distillation conclusions ...... 40
2.2 Modelling of short-path distillation ...... 41
2.2.1 Introduction ...... 41
2.2.2 Short-path literature ...... 41
2.2.3 Short-path conclusions ...... 47
2.3 Conclusions ...... 48
3 Modelling of RBD in tray columns 49
3.1 Tray column modelling ...... 49
3.1.1 Rigorous model ...... 50
3.1.2 Assumptions ...... 51
3.1.3 Initial conditions ...... 51
3.1.4 Integration ...... 52
3.1.5 Simplified m odel ...... 53
3.2 Comparison between simphfied and rigorous models ...... 53
3.2.1 Ethyl acetate case study ...... 53
3.2.2 Case study results ...... 56
3.3 Conclusions ...... 60 CONTENTS 6
4 Modelling of RBD in packed columns 64
4.1 Rate-based modelling of packed columns ...... 65
4.1.1 Modelling of mass and energy transfer ...... 65
4.1.2 Modelling of hydrodynamics ...... 66
4.1.3 Modelling of chemical reactions ...... 67
4.1.4 Packed column m odelling ...... 68
4.2 Reactive batch distillation case study ...... 69
4.2.1 Column design ...... 69
4.2.2 Column operation ...... 70
4.2.3 Effect of discretisation ...... 71
4.2.4 Determination of H EX ? ...... 72
4.3 Comparison between rate based and equilibrium m o d e ls ...... 80
4.4 Conclusion ...... 86
5 Control of reactive batch columns 88
5.1 In tro d u c tio n ...... 89
5.1.1 Control of batch distillation columns ...... 91
5.2 Controllability m ethods ...... 94
5.2.1 Simulation controllability analysis ...... 94
5.2.2 Frequency response controllability analysis ...... 97
5.3 Methods for controllability analysis ...... 102
5.3.1 Method for simulation controllability analysis ...... 102
5.3.2 Method for frequency response controllability analysis ...... 102
5.3.3 Robust method for linearisation ...... 103 CONTENTS 7
5.4 Controllability of batch distillation colum ns ...... 106
5.4.1 Case studies ...... 106
5.4.2 Scaling ...... 107
5.4.3 Linear models ...... 107
5.4.4 Frequency response based controllability ...... 110
5.4.5 Controller tuning ...... I l l
5.4.6 Non-linear model simulations ...... 112
5.5 Effect of reaction ...... 113
5.5.1 Simulation controllability ...... 114
5.5.2 Frequency response controllability ...... 114
5.6 Conclusions ...... 115
6 Modelling and control of short-path columns 124
6.1 In tro d u c tio n ...... 124
6.2 Modelling of short-path evaporators ...... 125
6.2.1 Modelling of him phenomena ...... 126
6.2.2 Modelling of evaporation phenomena ...... 128
6.3 Dynamic short-path distillation m o d e l ...... 130
6.3.1 Modelling assumptions ...... 131
6.3.2 Boundary and initial conditions ...... 134
6.3.3 Numerical solution ...... 134
6.4 Case study ...... 134
6.4.1 Effect of discretisation ...... 136
6.4.2 Effect of feed how rate ...... 137 CONTENTS 8
6.4.3 Effect of viscosity ...... 139
6.4.4 Effect of feed tem perature ...... 140
6.4.5 Effect of heat of reaction ...... 141
6.4.6 Effect of efficiency ...... 141
6.5 Control of short-path distillation ...... 142
6.5.1 Linearisation ...... 143
6.5.2 Frequency analysis ...... 143
6.5.3 Controlled response ...... 144
6.6 Conclusion ...... 145
7 Conclusions and directions for future work 149
7.1 C o n clu sio n s ...... 149
7.1.1 Modelling of reactive batch distillation in tray columns ...... 150
7.1.2 Modelling of reactive batch distillation in packed colum ns ...... 151
7.1.3 Control of reactive batch distillation ...... 152
7.1.4 Modelling and control of reactive short-path evaporators ...... 153
7.2 Directions for future work ...... 154
7.2.1 Model validation ...... 154
7.2.2 Modelling detail ...... 154
7.2.3 Further control studies ...... 155
Nomenclature 157
Bibliography 160 CONTENTS 9
A Process Models 166
A.l Equilibrium tray model ...... 166
A.2 Rate-based model of packing section ...... 169
A.3 Reboiler m odel ...... 173
A.4 Condenser m o d e l ...... 174
A.5 Reflux drum model ...... 175
A.6 Accumulator model ...... 177
B Linearisation and scaling methods 178
B.l Linearisation of model equations ...... 178
B.2 Scaling of the linear m odels ...... 179 List of Figures
1.1 Reactive batch distillation column ...... 18
1.2 W iped-Film evaporator ...... 20
1.3 Short-path e v a p o ra to r...... 21
3.1 Distillate composition (top) and distillate howrate (bottom) for the constant
rehux ratio study ...... 58
3.2 Accumulator composition (top) and holdup (bottom) for the constant rehux
ratio study ...... 59
3.3 Distillate composition (top) and distillate howrate (bottom) for controlled
composition stu d y ...... 60
3.4 Accumulator composition (top) and holdup (bottom) for controlled compo
sition stu d y ...... 61
3.5 Distillate how for non-reactive system ...... 62
3.6 Reboiler forward reaction rate and ethyl acetate composition (Rigorous
Model) ...... 62
3.7 Distillate composition (top) and distillate howrate (bottom) for controlled
composition study with reboiler heat input disturbance (bottom ) ...... 63
4.1 Mass and energy transfer between phases ...... 66
4.2 Vapour composition of Ethyl Acetate at 3 hrs (constant rehux ratio) . . . 74
10 LIST OF FIGURES 11
4.3 Composition and Distillate Flowrate - (TOP Constant reflux ratio - BOT
TOM Controlled Composition) ...... 77
4.4 HETP packing profile against time for Reactive Case Study (Constant reflux
ratio) ...... 78
4.5 Mean HETP profile for Reactive Case Study (Constant Reflux Ratio) . . . 78
4.6 HETP packing profile against time for Reactive Case Study (Controlled) . . 79
4.7 Mean HETP profile for Reactive Case Study (Controlled) ...... 79
4.8 Distillate composition (top) and distillate flowrate (bottom) for constant
reflux ratio policy ...... 84
4.9 Distillate composition (top) and distillate flowrate (bottom) for controlled composition policy ...... 84
4.10 Distillate composition (top), distillate flowrate (middle) and reboiler dis
turbance profile (bottom) for controlled composition policy ...... 85
5.1 Process for c o n tr o l ...... 89
5.2 Batch Column ...... 91
5.3 Frequency response of a first order system ...... 98
5.4 Block diagram of feedback control ...... 99
5.5 Controllability requirements ...... 101
5.6 Comparison of robust method(Simple) to standard method(Rigorous) . . . 104
5.7 Internal reflux ratio profiles ...... 106
5.8 Distillate composition response to step change in in p u ts ...... 109
5.9 Reboiler temperature response to step change in inputs ...... 110
5.10 Distillate composition response to unit step change in reflux flo w ...... 116
5.11 Tray column frequency response of distillate composition to reflux flow at
7.5 h r s ...... 117 LIST OF FIGURES 12
5.12 Packed column frequency response of distillate composition to reflux flow
at 7.5 h r s ...... 117
5.13 Frequency response of distillate composition to reflux flow for 10 tray column 118
5.14 Frequency response of distillate composition to reflux flow for 8m packed
co lu m n ...... 118
5.15 Tray column closed loop response to setpoint change (Linear model) .... 119
5.16 Packed column closed loop response to setpoint change (Linear model) . . 119
5.17 Magnitude composition response (10 tray column) ...... 120
5.18 Magnitude composition response (8m packed column) ...... 120
5.19 Composition controller error in response to 10% step increase in reboiler heat duty (Rigorous Tray Colum n) ...... 121
5.20 Composition controller error in response to 10% step increase in reboiler
heat duty (8m Packed Column) at 3 hours ...... 121
5.21 Distillate composition response of reactive and non-reactive tray columns . 122
5.22 Reboiler temperature response of reactive and non-reactive tray columns . . 122
5.23 Composition frequency response of reactive and non-reactive tray columns . 123
6.1 Short-path evaporator...... 126
6.2 Top view of mixing bow wave (adapted from McKenna 1995) ...... 127
6.3 Temperature profile in a wiped film evaporator (Lutisan et al 2002) ...... 128
6.4 Cross section of Evaporator ...... 131
6.5 Reaction Scheme ...... 135
6.6 Base case profiles: compositions (top), temperature (bottom ) ...... 136
6.7 Outlet Temperature profile resulting from disturbance (TOP) Steady-state
temperature profile (B ottom ) ...... 137 LIST OF FIGURES 13
6.8 Effect of feed flowrate on residence tim e ...... 138
6.9 Effect of feed flowrate on reactor yield ...... 139
6.10 Effect of film viscosity on reactor yield ...... 139
6.11 Effect of viscosity on residence time ...... 140
6.12 Effect of feed temperature on reactor yield ...... 140
6.13 Evaporator composition profile with no separation of V ...... 142
6.14 Product, B, composition step responses. Non-linear model (TOP), linear
model (BOTTOM) ...... 146
6.15 Short-path column frequency response of product B composition to feed flow 147
6.16 Short-path column frequency response of controller loop and temperature
disturbance ...... 147
6.17 Column composition response to a set-point change (controlled scaled linear
m o d e l) ...... 148
6.18 Column composition response to a feed temperature disturbance (controlled
scaled linear model) ...... 148
A.l Sieve tra y ...... 166
A.2 Packing section ...... 169
A.3 Reboiler...... 173
A.4 C o n d e n se r ...... 174
A.5 Reflux Drum ...... 176 List of Tables
2.1 Summary of papers on reactive batch distillation 1979 to present day .... 38
2.2 Summary of papers on reactive batch distillation 1979 to present day (con tinued) 39
2.3 Summary of papers on short-path distillation ...... 46
3.1 Column parameters ...... 56
3.2 Controller param eters ...... 57
3.3 Accumulator holdup and batch tim es ...... 63
4.1 Column dimensions and packing characteristics ...... 70
4.2 Comparison between level of discretisation for 8m packed colum n ...... 72
4.3 Comparison between modelling approaches. (EQ: equilibrium model, NEQ:
rate based model) ...... 83
5.1 Control S c h e m e s ...... 93
5.2 Inputs and outputs for linear model (6 Tray column at 3 hrs Un. point) . . 107
5.3 Controller tuning param eters ...... I l l
5.4 Integrated controller errors ...... 113
6.1 Short path column configuration ...... 135
14 C hapter 1
Introduction
This thesis is concerned with the modelling and control of reactive distillation in tray and packed batch columns as well as in short-path columns. Reactive
distillation offers advantages over separate reaction and separation steps, for
instance through improved yield, as volatile products are removed from the re action zone. However, by combining these processes, control is made more
difficult. In this chapter, reactive distillation is introduced with its advantages
and disadvantages. Then, more specifically, batch operation in packed and tray
columns and operation in short-path evaporators, is introduced. General com
ments are made about the control of these processes. The thesis motivations,
objectives and contributions are outlined. The outline of the rest of the thesis
is also presented.
1.1 Motivation
This thesis is concerned with the modelling and control of reactive distillation. Due to the complexity of combined reaction and separation and the operational constraints on these processes such as reaction temperature, they may be difficult to control in practice
(Sprensen and Skogestad, 1994). In this work, reactive distillation in tray and packed columns as well as short-path columns are considered. For batch columns, the changing
15 CHAPTER 1. INTRODUCTION 16
process conditions with time adds a extra dimension to the complexity of the control. For
the short-path column, typically used for the treatment of temperature sensitive products,
temperature control is extremely important. In order to study the control of these pro
cesses, it is necessary to develop rigorous dynamic models which accurately capture the
behaviour of the processes.
1.2 Reactive distillation
Reactive distillation is the combination of both reaction and separation into a single unit. This can offer particular advantages as reported in the literature by Taylor and Krishna
(2000), Doherty and Buzad (1992) and others:
• Combining two units into one can lead to significant capital savings
• Improved conversion of reactants, approaching 100% as volatile products are removed
from the reaction zone
• Low product concentrations in the column section, reducing unwanted side reactions
and leading to higher selectivity
• Azeotropes which would otherwise be formed by the reactants/products can be elim
inated through reaction
• Lower reboiler duty with exothermic reactions, as the heat of reaction assists in the
vaporisation
However, caution should also be taken when considering reactive distillation for the fol lowing reasons:
• The equilibrium enhancements offered by reactive distillation relies on one of the
products being the most volatile component in the system, so that it is preferentially removed from the reaction zone. CHAPTER 1. INTRODUCTION 17
• If the reaction is slow, it may be more economical to carry out the operation in
separate reaction and separation steps as a large column with large reflux wiU be
required with large capital and operating costs to achieve the required residence
time.
• The coupling of reaction and separation in one unit can result in a mismatch between
ideal process conditions. The optimal temperature and pressure for the reaction may
be very different to that for the separation. This is also an acute problem in packed
columns where the selection of the packing is often a compromise between separation
and reaction performance.
1.3 Batch column operation
In the manufacture of low volume, high value chemical products, and in situations where varying specifications of different products are required, the motivation for employing batch operations is well known. Reactive batch distillation can be carried out in a column such as that shown in Figure 1.1. Depending on the volatilities of the reactants compared to the products, the reaction may occur throughout the column or be confined to the reboiler. The separation section can be either a tray stack or a packing section. The choice depends on the operating conditions, although packed columns are particularly suitable if the reaction is to be heterogeneously catalysed in the packing. The reactants may be initially charged to the reboiler or one or more may be fed during semi-batch operation. A homogeneous catalyst, such as a concentrated mineral acid, may also be fed or charged initially. As with non-reactive distillation, reactive columns are normally operated at total reflux until a steady-state profile is developed before distillate withdrawal is started. Several production cuts may be made during the operation which may or may not be recycled. CHAPTER 1. INTRODUCTION 18
Condenser Reflux Drum
Reflux Distillate
Distillation Accumulator Column
iX M " Heat Supply
Reboiler
Figure 1.1: Reactive batch distillation column
1.3.1 Control of reactive batch distillation
The coupling of reaction and distillation processes creates a more complex process which is more difficult to control than either one on its own. In general, the use of automatic control systems in batch systems is fairly limited, there are a number of reasons for this:
• It is quite difficult, due to the transient nature of the process, to control the unit
based only on regulatory or tracking control of certain variables. The controllers
need to be adjusted according to the current state of the process with the aim of
achieving the desired state at the end of the batch.
• The desired state at the end of the batch and the performance of the process will
change with different charges and process requirements which further complicates
the procedures CHAPTER 1. INTRODUCTION 19
$ There can be difficulties in observing some state variables, e.g. composition, and if
they are to be controlled they may have to be inferred from quantities that are more
easily measured, e.g. temperature.
1.4 Short-path column operation
It is often not possible to heat many organic compounds to a temperature even close
to their normal boiling points without thermal decomposition occurring. The degree of
decomposition will also depend on the length of time the compounds are exposed to the heat source. Vacuum distillation enables the separation of these kinds of compounds by keeping the pressure and hence temperature low. Traditional batch distillation is some times unsuitable for vacuum distillation since the pressure drop across the column limits the amount of vacuum achievable in the still. A minimum pressure of around 50 mbars
(Erdweg, 1983) can be achieved under ideal conditions. Additionally, there are large res idence times within the still, which increases thermal decomposition. This has been the motivation for developing other distillation/ evaporation processes. These include falling film evaporators, wiped-film evaporators and short-path evaporators.
1.4.1 Falling film evaporators
Falling film evaporators are used successfully in many industries. The feed flows down heated walls forming a film. When operated under vacuum and high evaporation rates, a number of problems can occur. Hot spots form where material overheats resulting in decomposition. The laminar film restricts the distillation rates and large temperature difference can occur in the film. CH AFTER 1. INTRODUCTION 20
FEED VAPOUR TO EXTERNAL CONDENSER AND VACET'M PUMP
WIPING SYSTEM
RESIDUE
I'igure 1.2: Wiped-Filin evajiorator
1.4.2 Wiped film evaporators
In a wi])e(l-lilni evaporator, shown in Figure 1.2, the product is fed to the inside of a single tube and a inechaiiical, rotating, wiper spreads and moves the products, avoiding hot spots. The more volatile components generally run against the product flow, leaving the evaporator to an external condenser. The pressure drop between the evaporator and the external condenser determines the degree of vacuum achievable in this type of unit.
1.4.3 Short-path distillation
Short-path evaporator, shown in Figure 1.3, is in principle a wiped-film evaporator with an internal condenser. This eliminates the pressure drop associated with the pipework connecting the evaporator and the condenser. In tlieory, there is no pressure drop between the evaporating surface and the condensing surface because the distillation gap is of the same order of magnitude as the mean free path of the evajiorating molecules at the low operating pressure. 21
FEED
CONDENSER
HEATING JACKET
WTPING SA'STCM
— COOLING
RESIDUE DISTILLATE
Figure 1 .X: Siiori-jiatli evaporator
1.5 Objectives of this work
Reactive batch (listillatioii is an inlierently dynamic process since amount and composition of material within the column changes with time as the reaction proceeds and product is withdrawn. As a result, a dynamic mathematical model is required to describe its operation. Continuous distillation, sucli as short-path distillation, should be dynamically modelled if the model is to be used for control.
The objective of this thesis is to investigate the control of reactive distillation in batch columns, tray and packed, and in short path columns. Control has hitherto only been studied for reactive batch distillation in tray columns, and then only employing simplified dynamic models with linear tray dynamics. In order to study control fully, it is necessary to develop rigorous dynamic models that accurately capture the process behaviour. For batch processes in particular, process conditions change with time which will affect the controllabilitv and it is therefore essential that the conditions are accuratelv modelled. CHAPTER 1. INTRODUCTION 22
The higher the degree of rigour, the more accurately the process conditions and dynam ics are captured. However, more rigorous models are more computationally expensive to implement and they require more parametric data which may be unavailable or uncer tain. Additionally, more numerically complex models can be prone to numerical errors, introduced for instance during hnearisation.
Therefore, in this thesis, the degree of modeUing rigour required for both simulation and control purposes is explored in detail for tray and packed batch columns and short-path columns.
For batch tray columns, it is demonstrated that to accurately capture the change in process conditions, it is necessary to model pressure dynamics and employ a dynamic energy balance. For packed columns, distributed rate based modelhng is compared to lumped equilibrium modelling and it is found that due to the varying conditions within the packing, the efficiency changes, resulting in mismatch between the two methods. The short-path distillation column which has hitherto only been modelled at steady-state, is modelled using a dynamic rate based model, essential for investigating control.
Having developed the dynamic models, the control and controllability of these reactive distillation processes are examined. General control properties of reactive batch distillation are discussed and methods are presented for applying linear controUabihty tools to these non-linear process models. The linear models are then employed to demonstrate the implications for control when adopting one of the three processes.
1.6 Outline of this thesis
Chapter 2 presents a review of the literature on modelhng and control of reactive batch distillation in both tray and packed columns. The modelhng of short-path distillation columns is also considered. It is concluded that little work has been undertaken on the control of reactive batch distiUation in tray columns and no work has been undertaken on the control of reactive distillation in batch packed columns and short-path distillation columns. CHAPTER 1. INTRODUCTION 23
As already noted, rigorous dynamic models are required to enable the study of control.
Chapters 3 and 4 deal with the development of rigorous models to describe reactive dis tillation in batch tray and packed columns. The tray column is compared to a shghtly simplified, more numerically robust column for a number of different operating policies.
As packed columns are commonly modelled using equilibrium models, such as that for the tray column, the two modelling approaches are compared. In order to determine the equivalent tray column, the Height Equivalent to a Theoretical Plate (HETP) needs to be established and a method is presented for extracting this information from the column packing.
Chapter 5 is concerned with the control of tray and packed batch columns. In order to use linear control tools it is necessary to generate a linear approximation to the non-linear models. The rigorous tray column model proves to be numerically unstable for this purpose and an alternative scheme is presented for generating the necessary linear information from the simplified column model. The effect of the choice of column, packed or tray, the effect of reaction and the size of the column are investigated during the controllability study.
Chapter 6 considers the modelling and control of reactive distillation in short-path distilla tion columns. A dynamic short-path column is presented and used to investigate the effect of changes in operation on a complex, industrially motivated, reaction. The composition control of this process is also investigated.
Finally, in chapter 7, overall conclusions are drawn, and some possible directions for future work are outhned.
1.7 Main contributions
The main contributions of this thesis are that a rigorous, equilibrium, tray column model and a rigorous, rate-based, packed column model for reactive batch distillation are devel oped. A method is presented to determine the Height Equivalent to a Theoretical Plate
(HETP) from the packed column model and used to analyse how efficiency changes during operation. The rigorous tray column model proves to generate numerical difficulties when CHAPTER 1. INTRODUCTION 24
linearised. Therefore, a method is presented for coping with these difficulties by means of a simplified version of the model. A comparison is made between the controllability of packed and tray batch columns. Finally, a dynamic model for reactive distillation in a short-path column is presented and used to investigate the controllabhty of the process. C hapter 2
Literature review
In this chapter, the work that has previously been undertaken within the area of modelling and control of reactive hatch distillation and the modelling and
control of short-path distillation is reviewed. It is concluded that, while some
work on the control of batch tray columns has been done, the models used have
been simple. No work has been undertaken on the control of packed columns or
short-path columns. It is also noted that no work has been done on modelling
reaction in short-path columns.
2.1 Modelling of reactive batch distillation
2.1.1 Introduction
Reactive distillation can be operated in both continuous and batch modes of operation.
As is generally the case, continuous operation is best suited for large production volumes but, since the continuous design is tailored to a particular reaction/separation system, it lacks the flexibility afforded by batch operation. Control of both modes is complex but the batch mode offers additional challenges. In the continuous mode, control is generally required about a desired steady-state when not considering start-up or shut-down. Batch control, on the other hand, is inherently dynamic which adds an extra dimension to its
25 CHAPTER 2. LITERATURE REVIEW 26
complexity. The process is not controlled to a desired steady state but is controlled to a desired operating policy which may, and often does, change with time. The process itself changes with time, the volume in the reboiler is decreasing and the composition profile in the column changes. Consequently, the response of the process and its control system to disturbances wiU change with time. Efficient control is important, as minimising the impact of disturbances reduces batch inconsistencies and hence wastage. Good set-point tracking of an optimal operating policy ensures more economic operation. In this thesis, the focus is therefore on investigating the dynamic behaviour and the controllability of these processes as a more detailed understanding will lead to better control. In this section, the literature that has been published on reactive batch distillation (summarised in Tables 2.1 and 2.2) is examined to identify what work has been undertaken on the modelling and control of these processes.
2.1.2 Reactive batch distillation literature
Egly et al. (1979) developed a method for optimising the operating policies of batch distilla tion operations. The simple model used consisted of component mass and energy balances and a vapour-liquid equilibrium equation. Reaction could be considered throughout the liquid phase in the column and the still. The holdup of liquid in the trays and condenser were assumed to be constant and no pressure dynamics were considered. Optimisations were performed using a modified conjugated gradient method, minimising the batch time while maintaining product specifications in terms of yield and composition. Optimisations considered were constant reflux ratio, time variable reflux ratio and time variable reflux ratio with feed of a reactant. This was applied to a theoretical reactive case study with reaction confined to the still. The optimal, shortest batch time, was found for the time varying reflux ratio policy with feed of reactant which was 40% shorter than the constant reflux ratio policy. The authors also developed a non-linear, multi-variable control algo rithm that determined the required reflux ratio at any instant from temperatures in the column. CHAPTER 2. LITERATURE REVIEW 27
Cuille and Reklaitis (1986) considered the simulation of batch distillation with and with out liquid phase chemical reaction in the reboiler. The model was a system of differential and algebraic equations and the authors discussed the numerical problems associated with these systems and strategies for tackling these. The system was initialised at steady-state with total reflux and no reaction and integrated using Gear’s method. Constant volumet ric holdup on the trays was assumed and tray efficiencies were included. The reaction considered was an equilibrium estérification of 1-propanol and acetic acid. The rate of reaction was simplified by assuming no temperature dependence. However, the reaction is not practical for reactive batch distillation since 1-propanol, a reactant, is the most volatile component in the system and is hence removed preferentially. The authors com pared the non-reactive separation of cyclohexane and toluene with experimental data. The agreement was good for the distillate composition profiles for a number of constant reflux ratio simulations. However, there were no comparisons made for reactive batch distillation.
Reuter et al. (1989) considered the modelling of reactive distillation with control systems.
The model considered non-ideal stages and variable pressure. Assumptions included: con stant liquid hold-up on the trays and in the condenser and reaction only in the liquid phase.
They considered an equilibrium transestérification reaction, although no details were given on either the components or the reaction. The model was started from steady state, total reflux with no reaction and the steady state profile was calculated using Newton-Raphson procedures. The dynamic simulation was performed using a modified relaxation method.
The control scheme employed consisted of three controllers: Condenser cooling was used to control the condenser outlet temperature, reboiler steam flow for column pressure drop and distillate rate for the temperature at the top of the column. The authors compared both controlled and uncontrolled operation of the column but did not evaluate the two cases for disturbances. In the controlled case, the flow rate of the equilibrium limiting by-product was greater than that for the uncontrolled case, hence the reaction was faster.
Comparisons were made with experimental results and it was concluded that in practice. CHAPTER 2. LITERATURE REVIEW 28
the reflux ratio set by the controller in the simulation was always less than in the exper iment when maintaining a constant distillate composition. This suggest that the model agreement is not very good, although this is diflficult to verify from the results.
Sprensen and Skogestad (1994) investigated control strategies for reactive batch distil lation. They indicated that most authors had considered optimal control in order to maximise profit or minimise batch time. However, they argued that, in some cases, it may be a more important control objective to maintain product consistency between batches.
They considered the modelling of an industrial esteriflcation process where the reaction was limited to the reboiler. The reaction produced a separate polymer phase and water, the most volatile component in the reaction mixture. Modelling assumptions included: perfect mixing and equilibrium between the liquid and vapour phases, constant pressure, negligible vapour holdup, constant liquid enthalpies, linear tray hydraulics, total condensa tion without sub-cooling in the condenser, Raoult’s law for the liquid vapour equilibrium, perfectly controlled vapour holdup and immediate heat input. The authors highlighted the differences between non-reactive and reactive distillation by applying typical non-reactive operating policies to the reactive example. The open loop policies: constant reflux ra tio, R, and time varying reflux ratio, R(t), were found to be inappropriate, although there was acceptable separation, as the reactor temperature varied too greatly under dis turbances, resulting in varying polymer composition between batches. Only a constant product composition pohcy, implemented using feedback control, maintained the reactor within permissible temperature limits. The authors considered the controUabihty of the process by generating hnear models at different times during the batch. The linerised models were used to investigate the controllability by means of step responses and RGA analysis. They identified the reactor temperature and the distillate composition as being the most important variables to control. They also concluded that the same amount of side product being formed should be removed in the distillate at any time, that the system’s response varied with time as the conditions in the column/reactor change and that the trays had different and varying sensitivity to change and that the responses were generally CHAPTER 2. LITERATURE REVIEW 29
non-linear. The authors evaluated a series of control schemes: one point bottom control
(reactor temperature directly), two point control (controlling both distillate composition and reactor temperature) and one point column control (controlling the temperature on a tray in the column). It was concluded that the latter offered good control of the process with disturbances and did not have the disadvantages of the interactions encountered with two point control.
S0rensen et al. (1996) addressed the issues of optimal control of the same case study as
S0rensen and Skogestad (1994). They developed optimal profiles for the operating vari ables, assessed the controllability properties at optimal conditions, designed controllers to implement the optimal profiles and verified the stability and control performance. A series of optimisations were performed for maximum profit with and without raw material costs and for minimum batch time. The implementation was performed using the tem perature on a column tray to control reflux while maintaining the heat to the reboiler at the optimum value. The tray selected was the one that gave the largest response, identified in the controllability analysis. The stability of the controlled and uncontrolled model were assessed by introducing disturbances in the reboiler heat supply and in the reaction parameters which were used to indicate uncertainty in those parameters. The uncontrolled case gave significant variations in the reboiler temperature and breakthrough of the volatile reactant into the distillate. In the controlled case, the controllers were tuned using a linerised model about a series of operating points and a polynomial was used to describe the temperature set point profile. This yielded significant deviations in reboiler temperature only towards the end of the batch. The model was also interfaced to an industrial real-time control system and the controllers were implemented using the systems facilities. Rather than a polynomial description of the set points, a series of set points were used. This yielded good performance, similar to the continuous controller.
Mujtaba and Macchietto (1997) considered a computationally efficient method for optimis ing the operation of reactive batch distillation for maximum profit. They indicated that CHAPTER 2. LITERATURE REVIEW 30
this operational condition is strongly dependent on the cost parameters which in turn are dependent on market forces. They pointed out that to perform a rigorous optimisation procedure, involving integration of the model equations such as that used by Sqrensen et al. (1996), would be too computationally expensive to perform every time market con ditions change. The authors proposed a less computationally expensive method that used polynomial approximations to estimate the optimal operating policy from previously per formed optimisations. Their example was the esteriflcation of ethanol and ethanoic acid to produce ethyl acetate and water and they used simple models for the VLE and did not consider azeotropes such as that formed between ethyl ethanoate and ethanol. However, they indicated that the methodology is general and could be applied to more complex sys tems such as azeotropic mixtures by employing more rigorous Vapour-Liquid equilibrium models. The model includes reaction throughout the whole column in the liquid phase and assumes constant molar holdup on the plates and condenser. The energy balance is algebraic and hence assumes no change in liquid enthalpies. This is simpler than that used by Sprensen and Skogestad (1994) and Sprensen et al. (1996) as the tray holdup is assumed to be constant which is reasonable as they are not considering dynamics and the column would remain in pseudo steady-state during a large portion of the batch. The authors asserted that in reactive distillation where one of the products, desired or unde sired, is the most volatile component, finding the maximum yield is equivalent to finding the maximum production of distillate. This, they indicated from previous work, is equiv alent to the maximum profit for fixed batch time. A series of optimum constant reflux ratios were calculated for maximum product yield for a series of fixed batch times and for two product specifications using rigorous non-linear programming techniques (NLP). The results to these optimisations were used to develop polynomial descriptions of maximum conversion, optimum distillate, optimum reflux ratio and total reboiler heat load. These were all functions of batch time. These were then used in the formulation of a maximum profit optimisation which was an algebraic optimisation in only one variable, time. Opti misations performed using this method were approximately 200 times faster than using the rigorous method. They indicated that accuracy was very good and that this was mostly CHAPTER 2. LITERATURE REVIEW 31
due to the accuracy of the regression of the polynomials to the initial optimisation data
which behaved well, justifying their use of low order polynomials.
Wajge et al. (1997) examined the accuracy and speed of numerical methods for simulating
both reactive batch distillation and non-reactive distillation in packed columns. They in
dicated that packed columns models differ from tray model columns in that mass transfer effects need to be considered. They considered the finite difference technique which in volves converting the differential equations to algebraic equations of small intervals. This method was used for the simulation of batch distillation and it was concluded that the finite difference method is very computationally expensive. They considered an orthogonal collocation method where the equations are approximated to polynomials. They concluded that this was more efficient but the computational advantages over finite element methods were lost as the need for greater accuracy required the use of higher order polynomials.
They proposed a hybrid method called collocation on finite elements which permits high accuracy while retaining the use of low order polynomials and their shorter solution times.
They also identified that computation also took longer if the composition profiles became widely separated in the column. They indicated that the use of sparse matrix techniques in the solution also yielded improved solution times.
Wilson and Martinez (1997a) investigate methods for the estimation of state variables such as composition from temperature measurements in reactive batch distillation. The moti vation for this was that good composition control is essential in reactive batch distillation but the cost of on-line composition measurement is usually prohibitively expensive. The authors considered two types of state estimator. Firstly an Extended Luenberger Observer
(ELO) that derives its estimates from a linearised model of the process and secondly an
Extended Kalman Filter (EKF) which, although also based on a linear model, produces its estimates based on the statistical characteristics of the prevailing random process dis turbances and measurement noise. The two techniques were compared for an industrial multicomponent reaction. Two models were used, a very simple binary distillation model CHAPTER 2. LITERATURE REVIEW 32
and a slightly more complex multicomponent distillation model. The simple non-hnear
model was used to generate the Hnear models for the state observers as the more complex,
multi-component model was too computationaUy expensive for this purpose. Tempera ture measurements were taken from the more detailed model and measurement noise was added. The authors demonstrated that the EKF estimator produced better estimates that the ELO, which had stability problems. They concluded that the process mismatch between the models did result in reduced estimator accuracy and further refinement was required. However, they concluded that this accuracy was sufficient basis for composition control.
Wilson and Martinez (1997b) considered the automation of batch processes using fuzzy modelling and reinforcement learning and applied their techniques to reactive batch dis tillation. They stated that, in general, the operation of batch processes rehed heavily on the skills of operators to achieve the desired product due to the difficulties of developing effective automatic control systems. They suggested that the abilities of the operator can be imitated, and even improved on, by the use of embedded autonomous agents. The agent “perceives” the state of the process at each time step, executes an action and re ceives a reward or payoff in return. The agent’s task is to react continuously to the process state, influenced by disturbances and events, by determining a sequence of actions which maximises some cumulative measure of rewards, driving the process towards its “goal state”. Wilson and Martinez (1997b) concluded that the most common algorithm for this reinforcement learning, Q-Learning, had significant stabiHty problems and computational expense when appHed to batch processes where states vary with time. They presented a hybrid approach where Q-Learning was combined with fuzzy rules relating process states to control actions. This hybrid approach was termed fuzzy Q-Learning. This technique was successfully demonstrated for the control of the industrial reactive distiUation case study presented in Wilson and Martinez (1997a).
Wajge and Reklaitis (1998) presented a methodology for campaign optimisation of reactive batch distillations. The authors considered the compositions and amounts of off-cuts as CHAPTER 2. LITERATURE REVIEW 33
optimisation variables in an upper level optimisation with reflux ratio profile optimisation
as a lower level decision problem. To simplify the procedure, they introduced the concept of
Distillation Characteristics which, is the composition profile developed at total reflux. They
deduced that for reversible reactions where holdup in the column is neghgible, two mixtures
have the same distillation characteristic if the products and reactants are in the same stoichiometric proportions. Thus the optimal operational policy for the column would be the same in both cases. They considered the recycling of off-cuts, mixed with fresh feed, which have the same distillation characteristic as the feed charge. The authors illustrated their methodology with the esteriflcation of ethanol, using a packed column based on the model reported by Wajge et al. (1997). They concluded that the distillation characteristic is relevant in the design of campaign structures which minimise off-cut fragmentation and reduce operational complexity by minimising the number of distinct distillation tasks. In addition to a better reprocessing strategy, it offers insight into the trade-off between production rate, reactant utiUsation and waste generation. The authors noted that the feature of distillation characteristics breaks down if the reaction is irreversible and if the holdup within the column is not negligible. This would Hmit its appHcability to real processes.
Bollyn and Wright (1998) considered the use of experimental data in developing and refining a dynamic model describing a fed-batch reactive distillation column. The reaction considered was the synthesis of ethyl esters of pentenoic acid by substitution of aUyl alcohol for ethanol on triethyl orthoacetate. The reaction was assumed to occur in the reboiler only. They simphfled the reaction scheme by ignoring reaction steps that occurred sufficiently fast to be considered instantaneous, but were stiU left with 3 equilibrium reactions and 4 ehminations, yielding a total of 10 reactions. Their objective was to establish an optimal operating policy, through simulation, such that a high conversion of triethyl acetate and a high selectivity for the desired ethyl ester was achieved while minimising excess alcohol. They also noted that in previous work, substantial time was spent ensuring that the vapour-liquid equilibrium models were accurate and that less effort CHAPTER 2. LITERATURE REVIEW 34
was spent on the details of the reaction kinetics. They indicated that more attention
should be paid to the kinetics as they are more temperature dependent than most other
physical properties. Few details of the modeUing assumptions were given except that the
process was simulated in BatchCAD which uses a rigorous dynamic mass transfer based
distillation model. Experimental data was collected and analysed in order to refine the models. This was carried out at several levels, starting with a lab batch investigation into the kinetics moving through to pilot plant. The refined models were used at each stage to enable effective targeting of further experiments to enhance the model. This led to the development of an optimal operating policy which was successfully implemented on the pilot plant resulting in an improvement in selectivity from about 50% to over 98%.
Li et al. (1998) considered the optimisation of a semi-batch distillation process and used an industrial process to validate their model. The reaction considered was an industrial transestérification reaction where an alcohol product is the most volatile reactant. A semi-detailed model was developed including constant holdup, constant tray efficiencies and constant pressure profile. The pressure on each tray was interpolated from the top and bottom pressures in the industrial column and the Murphree tray efficiency was de duced by trial and error through comparison to the experimental results. The simulation results were sufficiently close to the experimental to justify the use of the model for op timisation. The optimisation was performed for minimum batch time employing control vector parameterisation (CVP) and sequential quadratic programming. They optimised the feed howrate of the alcohol, the reflux ratio and the switching time between the main and off-cuts. Two scenarios were considered: firstly the optimisation of the process under present requirements, where a 30% time saving was achieved, and the optimisation of the process under slightly lower product purity requirements. The authors acknowledged that although their solutions were feasible, it is hkely that sub-optimal solutions were found due to the nonconvexity and complexity of the problem.
Xu and Dudukovic (1999) considered the modelhng of a photo reaction in a semi-batch CHAPTER 2. LITERATURE REVIEW 35
reactive distillation column. In their model, they considered reaction in both liquid and
vapour phases and indicated that this was important for photo reactions because differ
ent kinetic behaviours occur in the two phases. Their model was a staged equilibrium
model where both liquid and vapour phases are considered to be at equilibrium in each
compartment. They included constant volumetric holdup of each phase, constant pres sure and assumed perfect mixing. The reaction considered was the chlorination of toluene where selectivity for the desired benzyl chloride is enhanced by reactive distillation as the product is distilled away from the reaction zone. The presence of UV light also enhances the selectivity as it promotes the desired chlorination of the methyl group and not the chlorination of the benzene ring. They performed simulations for a series of three different column configurations and compared it with experimental data. The comparison with experimental data was poor and it was concluded that the simulation only agreed with the experimental data in terms of trends. It was only as the real system approached ideal operating conditions described by the model that the best performance could be achieved.
Venimadhavan et al. (1999) considered the synthesis of a reactive batch distillation pro cess for the manufacture of butyl acetate, an important industrial solvent. The reaction is an esteriflcation of butanol and acetic acid to form butyl acetate and water. They de veloped a very simple model to capture the essence of the process and to provide insight for exploring process alternatives. Assumptions included: liquid-phase reaction confined to the reboiler, constant molar overflow, operating reflux sufficiently large to be approx imated as total reflux for the purposes of calculating column profiles, and tray holdups negligible compared with that of the still. They considered a reflux policy such that the instantaneous Damkohler number remained approximately constant which is equivalent to the rate of product removal being kept equal to the rate of production. The reaction kinetic parameters were regressed from earher published work. The phase equilibrium is quite complex with organic and aqueous phases being formed. The authors indicated that there was a debate within literature as to which of two azeotropes was the lightest boiling azeotrope: ternary, butanol-water-butyl acetate or binary, water-butyl acetate. CHAPTER 2. LITERATURE REVIEW 36
Using topological arguments for the nature of singular points on the residue curve map for
the non-reacting ternary system with acetic acid absent, they concluded that the lightest
was the ternary azeotrope. They indicated that despite the two azeotropes having close
boiling temperatures, the detection of the correct azeotrope had large implications for the
process design. In this case, the aqueous phase, containing a small amount of butanol,
was removed as distillate and the organic phase was returned as reflux. This results in the
exclusion of water and the accumulation of butyl acetate in the still which can be purifled
through non-reactive distillation at the end of the reaction. A second model was proposed
which included constant holdup on the trays. This model was compared favourably with
the simplifled model although the inclusion of traydynamics resulted in a slower response.
Monroy-Lopereba and Alvarez-Ramirez (2000) commented that optimisation approaches, such as that presented by Mujtaba and Macchietto (1997) have an important drawback in that the optimal solution depends strongly on the model and model parameters and there fore feedback control is essential in order to maintain optimum profitability in the wake of uncertainty. The author’s objective was to obtain an output-feedback controller with guaranteed tracking properties, despite uncertainties in the dynamics of the RED process.
They also wanted to demonstrate that the resulting reflux ratio policy approaches that obtained via optimisation techniques. The controller design is based on an approximate model of the composition dynamics and makes use of a reduced order observer to estimate the modelhng error. The resulting controUer is shown to have the same structure as a PID controUer with anti-reset windup. The controller performance was tested on the column model as presented by Mujtaba and Macchietto (1997) and demonstrated that the result ing reflux ratio pohcy approached the optimal.
Schneider et al. (2001) developed a rate based model for reactive distiUation in a packed distiUation column. The heterogeniously catalysed reaction was assumed to be pseudo- homogenious. The column model contained dynamic mass and energy balances. The
MaxweU-Stefan equations were used to describe the interfacial mass transfer within the CHAPTER 2. LITERATURE REVIEW 37
structured packing. The liquid holdup in the packing was determined using an experi mentally derived correlation, specific to the packing. The authors neglected vapour phase holdup due to the low pressure (< 1.2 Bar). No details on the discretisation method em ployed were given. The model was validated with experimental results from the synthesis of methyl acetate in a semi-batch column. Following a sensitivity analysis it was concluded that the reaction kinetics and models of the column periphery have a significant influence on the simulation results. It was indicated that, for the column to be used for control purposes, some form of model reduction would be necessary. g I to A u th ors R ea ctio n Model Liquid P ressu re W ork location D yn am ics D yn am ics Egly et al. (1979) liquid equilibrium constant constant optimisation of reflux ratio phase molar holdup policies I H Cuille and Reklaitis (1986) liquid equilibrium constant constant simulation only phase + efficiency volume holdup I Reuter et al. (1989) liquid equilibrium constant variable pressure and temperature control phase + efficiency volume holdup g Sprensen and Skogestad (1994) reboiler equilibrium linear constant controllability and control strategies only Sprensen et al. (1996) reboiler equilibrium linear constant optimisation and implementation only of optimal policies Mujtaba and Macchietto (1997) liquid equilibrium constant constant methods for online optimisation phase molar holdup Wajge et al. (1997) bquid rate-based variable from variable from accuracy and efficiency of spatial phase packed correlations correlations discretisation techniques Wilson and Martinez (1997a) reboiler equilibrium constant constant state estimation of composition only molar holdup from column temperatures Wilson and Martinez (1997b) reboiler equilibrium unknown unknown application of neuro-networks only to control
Table 2.1; Summary of papers on reactive batch distillation 1979 to present day
CO oo i
(\2
A u th ors R eaction M od el Liquid P ressu re W ork location D yn am ics D yn am ics I Wajge and Reklaitis (1998) hquid rate-based variable from variable from campaign optimisation phase packed correlations correlations I Bollyn and Wright (1998) liquid rate-based unknown unknown optimisation and phase packed model validation g Li et al. (1998) reboiler equihbrium constant variable optimisation and only + efficiency molar holdup model validation Xu and Dudukovic (1999) liquid and equilibrium constant constant simulation only vapour molar holdup Venimadhavan et al. (1999) reboiler equilibrium constant constant development of novel only molar holdup operating policy Monroy-Lopereba and Alvarez-Ramirez (2000) liquid equilibrium constant constant Model based control phase molar holdup Schneider et al. (2001) hquid rate-based variable from constant Simulation and phase packed correlations model validation
Table 2.2: Summary of papers on reactive batch distillation 1979 to present day (continued)
CO CO CHAPTER 2. LITERATURE REVIEW 40
2.1.3 Reactive batch distillation conclusions
Most of the work on reactive batch distillation has focussed on modelling of reactive batch
distillation with some work on optimisation of operating policies and some limited work on
control. The level of modelhng detail has varied but most authors have used equihbrium
modelling with either constant hquid holdup on the trays or hnearised tray hydrauhcs.
Little consideration has been given to pressure dynamics with vapour holdup neglected
and pressure assumed constant.
Few papers have considered control, (Reuter et al. (1989), Wilson and Martinez (1997a),
Wilson and Martinez (1997b) and Monroy-Lopereba and Alvarez-Ramirez (2000)) but mostly for simple models. It is also noted, that with the exception of Sqrensen and Skoges tad (1994) and Sqrensen et al. (1996), no work has been undertaken on the controUabihty of reactive batch distillation although analysis of controUabihty forms the basis for under standing the features that make control of these processes difficult. The fiexibihty which is offered by batch operations would be enhanced by better knowledge of how to modify the process, its operation or control structure, to yield better controUer performance.
Note that reactive distiUation in packed columns has rarely been considered in packed columns, those that have (Wajge et al. (1997), Wajge and Reklaitis (1998), Bollyn and
Wright (1998) and Schneider et al. (2001)), have importantly not considered the vapour phase which is essential for considering pressure dynamics.
It is therefore the objective of this thesis to investigate in more detail the controUabihty and control of batch reactive distiUation in general. In particular, it is felt that a more detailed equilibrium model of the batch distiUation column, in particular the inclusion of pressure dynamics for the study, would give greater insight into the process behaviour.
An investigation into controUabihty of reactive distiUation in batch packed columns is also proposed, not previously undertaken in any form. Conclusion wiU be drawn which may influence the selection of separation medium, trays or packing, during design. CHAPTER 2. LITERATURE REVIEW 41
2.2 Modelling of short-path distillation
2.2.1 Introduction
Short-path distillation is considered in this section and emphasis has been placed on papers
that consider the modelling of these processes. As noted earlier, short-path, or molecular,
distiUation occurs at very low pressures. Short-path evaporators are ideal for processing
temperature sensitive materials as much higher levels of vacuum can be achieved than in
a normal batch column due to the absence of trays. Also the residence time of material
in the column is much lower. A summary of the literature is given in Table 2.3.
2.2.2 Short-path literature
Cvengros et al. (1995) experimentally evaluated the residence time distribution (RTD)
curves obtained in the hquid film formed on the surface of the wiped film evaporator.
The experimental apparatus was a short-path evaporator with a segmented wiper in the
absence of distiUation. The experimental Uquid was Triethylene glycol (TEG) and an
aqueous solution of NaCl was used as a trace. Residence time was measured via a conduc
tivity probe. The wiping process was modeUed by a cascade of apparatuses with laminar falling film ideaUy mixed at the exit from each stage in a mixer with zero residence time.
The authors compared the effect of the Uquid load and Uquid viscosity in both the experi mental apparatus and the model. It was concluded that the model agreed closely with the experimental behaviour and provided a method for estimating the efficiency of the wiping in a real system.
Lutisan and Cvengros (1995a) indicated that in order to achieve molecular distiUation, the size of the gap separating the evaporating and condensing surfaces should be less than, or at least comparable to, the mean free path of the molecules at the prevailing pressure. This would ensure efficient evaporation. They indicated that the mean free path predicted by the kinetic theory of ideal gases gave a significantly smaller mean free CHAPTER 2. LITERATURE REVIEW 42
path than experience suggested. They gave an example that predicted a mean free path
of 1 mm but in practice, the distillation rate did not drop for separations up to 50 mm.
They concluded that the prediction of the mean free path must also depend on other
factors such as geometry. In order to get a better prediction, they developed a one
dimensional model for molecular distillation based on Direct Simulation Monte Carlo
method (DSMC) which is used to determine particle velocities within the distillation space.
The data was used to calculate mean free path and other macroscopic variables such as
particle density, collision frequency and kinetic temperature throughout the distillation
space. These phenomena were compared for different separation spaces, and condenser
and evaporator temperatures. The model was also used to compare the efficiency of
molecular distillation from concave and convex surfaces. The efficiency was defined as the ratio of the actual diffusion rate to the ideal, Langmuir-Knudsen diffusion. They indicated that the efficiency was lowest for concave surfaces, particularly at small diameters where
the curvature is highest. They also indicated that the efficiency was higher for concave
surfaces because the space above expands rapidly so density falls.
Lutisan and Cvengros (1995b) used the one dimensional model of a short path molecular
distiller, (Lutisan and Cvengros, 1995a), to investigate the effect of inert gas pressure on the molecular distillation process. They concluded that the presence of the inert gas at lower partial pressures than the distilling components has negligible effect but at higher partial pressure, the effect was quite marked, significantly reducing the distiUation rate.
McKenna (1995) developed a model for the design of a wiped film evaporator for the separation of volatile components from polymer solutions. It was noted that for highly viscous systems, the action of gravity might not be sufficient to induce a reasonable flow along the evaporator. The movement of Uquid can be enhanced by inclining the normaUy vertical wiper blades providing a pumping action. The mixing of the Uquid film was considered in detail as well as mass transfer through the Uquid film. The resistance to mass transfer at the surface between vapour and film was considered to be relatively CHAPTER 2. LITERATURE REVIEW 43
negligible and therefore at equilibrium. The model was used to examine the separation
performance and this was found to increase with the speed at which the wiper blades
remix the film. However, the author noted that there appears to be a limiting rotational
speed, above which significant gains in mass transfer are obtained only at the expense of
very large increases in power consumption. The results from the model were compared
with data pubHshed on commercially available wiped film evaporators and were found to
agree well, both quabtatively and quantitatively.
Lutisan et al. (1998) considered the inclusion of an entrainment separator, a sieve inserted
into the distiUation gap to tackle the problem of entrainment. When the feed enters the
column, there is a rapid evaporation and escape of dissolved gases and low volatiUty sol vents which can cause splashing. Material on the evaporator surface can be transported
into the distillate stream, causing a drop in efficiency. This problem is particularly acute
where the distillation gap is very small. A sieve, inserted between the evaporating and
condensing surfaces, traps both entrained Uquid and evaporated solvents and gases. Re
evaporation of the solvents occurs on the other side of sieve. The Uquid film was modelled
along the same lines as the author’s earlier paper (Micov et al., 1997) and the vapour phase was modelled using a direct simulation Monte Carlo method. The results obtained
indicated that the sieve improves the composition of the distiUate but decreases the dis
tillation rate.
BatisteUa et al. (2000) considered the implementation of the non-ideal vapour phase model
developed by Lutisan and Cvengros (1995b) in their DISMOL software which was com bined with work on the Uquid phase reported in BatisteUa and Maciel (1996). They indicated that this software would enable non-ideal systems to be studied. The authors indicated that the model would support multicomponent systems although only a single component was considered in their case study. The model was employed to determine how efUciency is affected by system pressure, condenser temperatures, separation distance between evaporator and condenser for different layouts. It is noted that for the conditions CHAPTER 2. LITERATURE REVIEW 44
investigated the efficiency did not drop below 60 %.
Cvengros et al. (2000a) indicated that the film surface temperature is an important factor
in determining the efficiency of a falling film evaporator. In their paper, they developed
a model to determine how the surface temperature profile varies with feed temperature
and liquid load. The equipment modelled was a falling film evaporator (unwiped) and the
single component film was assumed to be laminar, modelled by the Nusselt equation. Heat
transfer was considered axially and radially within the film. Resistance in the vapour phase
was neglected and the evaporation rate was assumed to be governed by the Langmuir-
Knudsen equation. The film surface temperature rises along the length of the evaporator
until a steady-state temperature is reached. The authors concluded that the feed should ideally be preheated to this temperature to avoid using a portion of the evaporator to
heat the feed. During this post-heat portion of the evaporator, evaporation rate is lower
than at the steady-state temperature and hence separation efficiency is lower. They also
compared the film temperatures calculated by the “non-approximate” model presented in
this paper with the linear development of temperature presented in their earlier paper
(Micov et ah, 1997) and concluded that the linear assumption gave a good approximation to the surface temperature.
Cvengros et al. (2000b) investigated the modelling of fractionation in a molecular evap orator. The equipment modelled was a falling film evaporator (unwiped) with a divided condenser which allows product to be withdrawn at various heights thereby permitting fractionation. The model was based on the one developed in their previous paper (Micov et ah, 1997) which was extended to model the behaviour of the liquid film created on the condenser surface. They demonstrated that with properly adjusted process parameters, fractions with different compositions can be obtained. The condenser at the top of the column will be richest in the more volatile component, the lowest richest in the less volatile component. They also commented that the divided condenser is generally more efficient due to the lower liquid load as a result of the side drainage. The model was compared to CHAPTER 2. LITERATURE REVIEW 45
experimental results showing a reasonably good agreement.
Kawala and Dakiniewicz (2002) developed a model to describe high vacuum distillation
in an evaporator with rotating discs. The rotating discs are arranged perpendicularly to
the flow of liquid providing a greater area for evaporation than conventional wiped film
evaporators. The model agreed well with experimental data and they concluded that the
rate of evaporation is highly dependent on the size of the vapour outlet and that this
should be as large as possible.
Lutisan et al. (2002) indicated that a real wiped film evaporator operates between two limits. At the lower, laminar, limit there are large concentration gradients between the
outer, heated, surface and the inner surface where the volatile components evaporate. At
the upper limit, due to wiping, the concentration and temperature gradients are elimi nated. The authors, using models developed earlier (Cvengros et ah, 2000a), explored the
differences between these two regimes. They assumed that the pressure was sufficiently low to neglect vapour phase modelling. They demonstrated that the distillation rate in the turbulent regime is much higher than in the laminar regime at the same surface tem perature of the evaporating cylinder and that consequently, due to the lower residence time for a given distillate rate, thermal decomposition would be smaller. They noted that the regime had minimal effect on the evaporator separation efficiency. They compared the experimental values of relative volatility with the model showing qualitative agreement between experiments and model. I
A u th ors E vaporator Q uantity Liquid Vapour Phase W ork toI T y p e V ariation B ehaviour M od el Cvengros et al. (1995) wiped-hlm radial lam inar N/A Residence time distribution -1- axial + mixing section Lutisan and Cvengros (1995a) N/A radial N/A DSMC Effect of mean free path Hi only on efhciency Lutisan and Cvengros (1995b) N/A radial N /A DSMC Effect of inert gases i on efficiency McKenna (1995) wiped-hlm radial complex Equilibrium Effect of mixing I Lutisan et al. (1998) N/A radial N /A DSMC Separation efhciency
BatisteUa et al. (2000) falUng-hlm radial laminar DSMC Effect of pressure and condenser temp on efhciency. Cvengros et al. (2000a) falling-hlm radial lam inar Langmuir Effect of feed temperature + axial -Knudsen Cvengros et al. (2000b) faUing-hlm radial lam inar Langmuir Fractionation with -|- axial -Knudsen divided condenser Kawala and Dakiniewicz (2002) rotating N/A N/A Langmuir- Simulation and discs Knudsen model vahdation Lutisan et al. (2002) wiped-hlm axial lam inar Langmuir Laminar and turbulent (radial) 4- turbulent Knudsen regimes effect on efhciency
Table 2.3: Summary of papers on short-path distillation
4^ CHAPTER 2. LITERATURE REVIEW 47
2.2.3 Short-path conclusions
As can be seen from the literature review on short-path distillation, the modeUing of
short-path distiUation can be broken into two: the treatment of the liquid phase, both the
evaporator film and the condenser film, and the treatment of the vapour phase and the
resistance it offers to mass transfer.
The liquid film may be treated as laminar, for example in falling film evaporators, where
large concentration and temperature profiles are to be expected radially through the depth
of the film (e.g. Lutisan et al. (2002)). During wiping the film can be considered to be weU mixed without a radial profile. McKenna (1995) for example, considered the modeUing
of the mixing process in detail. Lutisan et al. (2002) indicated that the mixing produces
higher distiUation rates but not significantly higher efficiency. Regardless of the Uquid flow regime, variations are expected in temperature and composition along the axial length of the evaporator.
The vapour phase resistance can be ignored at low pressures when the distiUation gap, or the distance between the evaporator film surface and the condenser film surface, is of the same order of magnitude as the mean free path of the gas molecules. Here, the rate of evaporation is governed solely by the Langmuir-Knudsen equation for molecular evaporation. Otherwise, some resistance is given by the vapour, reducing the distiUation rate. This can be modeUed by solution of the MaxweU-Stefan equations and authors have tended to adopt a direct simulation Monte-Carlo method (DSMC) to achieve this (Lutisan and Cvengros (1995a)(1995b) and BatisteUa et al. (2000)). The efficiency of the system is also effected by re-evaporation of material from the condenser. TypicaUy, lowering the temperature of the condenser has the effect of reducing this effect. It is noted, however, that none of the authors have considered carrying out a reaction within the evaporator film although, this is done industriaUy. Also, all of the authors have considered steady- state models, which is reasonable as these units tend to be operated on a continuous basis. CHAPTER 2. LITERATURE REVIEW 48
However, these models cannot be used to consider the control of these processes where
a dynamic model is required. It is therefore necessary to develop a dynamic model that
considered a possible reaction in order to investigate the control and controllability of
these processes.
2.3 Conclusions
In this chapter, the state of the art of reactive batch distiUation modelhng and control
has been presented together with that for the modelling of short-path evaporators. In the following chapters, rigorous dynamic models of reactive distiUation in tray and packed
batch columns and in short-path columns will be presented that wiU subsequently be used for control and controllability studies in the final chapters of this thesis. C hapter 3
Modelling of reactive batch distillation in tray columns
In this chapter, the modelling of batch tray columns with chemical reaction in
the liquid phase is considered. The features and assumptions of both a rigorous
model and a simplified model are presented. A case study is presented and the
two models are compared for tioo different operating policies, a constant reflux
operating policy and a controlled distillate composition study. The models are
also compared under reboiler disturbances for the controlled case study. It is
concluded that the simplified model is significantly different to the rigorous, es
pecially under varying process conditions. Therefore the rigorous model should
be adopted when simulating reactive batch distillation. However, the rigorous
model has a number of disadvantages: The initialisation of the model is more
complicated and the computational expense is high.
3,1 Tray column modelling
In this chapter, two process models are presented for the simulation of reactive batch distillation in tray columns. Firstly, a rigorous model which considers the modelling
49 CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 50
of both liquid and vapour phases with dynamic mass and energy balances is presented.
Secondly, a simplified model, although more rigorous than models previously used for control studies, neglecting pressure dynamics and with an algebraic energy balance is considered.
3.1.1 Rigorous model
The hterature review in chapter 2 indicated that simple dynamic models have been used for studying the control of reactive batch distillation in tray columns. In this thesis, a more rigorous equilibrium tray model is developed and used with the following features:
• Incorporates a dynamic energy balance equation instead of an algebraic energy bal ance where liquid enthalpy on the tray is assumed constant or constant molar over
flow where the energy balance is neglected
• Both liquid and vapour phase tray holdups are considered, their combined value
being a function of the prevailing pressure and the inter-tray spacing
• Liquid tray holdup is allowed to vary, the liquid flowrate from the tray being deter
mined by the Francis weir formula
• A detailed pressure drop equation that takes account of both dry and wet head loss
on each tray, thereby determining the vapour flowrate
• Rigorous vapour-hquid equilibria, considering non-ideality in both liquid and vapour
phases can be implemented if necessary
• Condenser can operate as partial, total or subcooled
• Reaction is considered throughout the hquid phase and is incorporated into the
component mole balances in the reboiler, trays, reflux drum and accumulator. Heat
of reaction, where applicable, is incorporated into the energy balance. Reaction is
rate based CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 51
Dynamic material and energy balances are also used to model the accumulator, reflux
drum and reboiler drum. In each of these, both Hquid and vapour holdups are taken into
account.
3.1.2 Assumptions
The assumptions for the rigorous model are:
1. No entrainment effects
2. No downcomer dynamics
3. Adiabatic operation
4. Phase equihbrium
5. Perfect mixing
6. Immediate heat input
7. Negligible holdup in the condenser
Murphree plate efficiencies can be introduced to take account of imperfect equihbrium
although it is important to note that no methods have been reported in the Hterature on how to account for chemical reactions in efficiency calculations (Ruiz et ah, 1995). It is also worth noting that the perfect mixing assumption is one of the more restrictive in reactive distiUation since it in effect suggests that each tray is a completely mixed stirred tank reactor. The model equations are given in Appendix A.
3.1.3 Initial conditions
The mathematical model of a batch reactive tray distiUation column as used in this work
(Appendix A) forms a set of differential and algebraic equations (DAEs). For integration of the DAE system, a consistent set of initial conditions is required. The system of DAEs CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 52
has an index of one and therefore the number of initial conditions required is equal to the
number of differential variables or states. Each sub-model contains (NcPl) states, arising
from the Nc dynamic component balances plus a dynamic energy balance. A practical
initial set of specifications would be the mole fractions of Nc — 1 components in the liquid
phase, the total liquid holdup and the temperature in each unit.
An alternative, adopted in this work, is to make the Nc — 1 component specifications
and liquid holdup specifications but instead of specifying the temperature on all trays and
reboiler, a single temperature specification is made on the top tray and the pressure drop
between each tray and the one below is instead initialised. The practical advantage of
doing this is that the initial conditions are more flexible. The initial tray temperatures are strong functions of pressure and composition. However, the pressure drops between
trays are much weaker functions of pressure and composition and are instead strongly
dependent on the height of liquid on the tray and the reboiler heat input. Therefore, the second method is more readily transferable to different mixtures and different operating
pressures with only one temperature specification required in the column.
3.1.4 Integration
Having determined a set of consistent initial conditions as outlined above, it is then nec essary to solve the DAE system. This has to be done numerically due to the non hnearity of the equation system. The integration of the DAE system describing batch reactive distillation, as with most process engineering apphcations can be stiff due to phenomena operating on widely different time scales.
Implicit numerical integration techniques are better suited than explicit ones for stiff sys tems as they have superior stability properties. The process modelling software pPROMS
(Process Systems Enterprises Ltd., 1999), employed in this work, uses the backward dif ference formula (BDF - Gear (1971)) family of methods. In the BDF method, the order CHAPTER 3. MODELLING DE RBD IN TRAY COLUMNS 53
of the integration and the size of time step are varied automatically to ensure that the
longest possible time steps are taken while satisfying the error tolerances of the user.
3.1.5 Simplified model
The simphhed model employed in this thesis is closely based on the rigorous model. It is
more numerically robust for the purposes of linearisation, as discussed in Chapter 5. The
following assumptions are made:
• The variable pressure feature is replaced by a constant pressure drop across the
column stages and constant pressure in the condenser
• The dynamic energy balances on trays, reboiler and reflux drum are replaced by
algebraic energy balances. The assumption is made that the enthalpy of the liquid phase remains constant.
• Vapour phase holdup is assumed to be neghgible
For the simplified model, the initial condition specifications are the total material holdups
and the Nc — 1 composition specifications.
3.2 Comparison between simplified and rigorous models
Having presented the rigorous and simplified models it is necessary to compare the two approaches. This is in order to justify the adoption of the rigorous model over the sim- phfied model for simulations. Two operating policies: constant reflux ratio and controlled distillate composition are used for the production of a 0.6 mole fraction mixture of ethyl acetate.
3.2.1 Ethyl acetate case study
In order to compare the simplified model with the rigorous model, the production of ethyl acetate is considered. The reaction is an equilibrium estérification of ethanol (Tj, = 352 K) CHAPTER 3. MODELLING OE RBD IN TRAY COLUMNS 54
and acetic acid (Ti = 391 K) which forms the ester, ethyl acetate (Tb = 350 K) and water
(Tb = 373 A").
Acetic Acld(l) + Ethanol(2) ^ Ethyl Acetate(S) + Water(4)
As can be seen, the boiling point of ethyl acetate is lower than any other component
in the mixture, making it suitable for reactive batch distillation. The product ester will
be separated from the reactants preferentially through the distillation as it is produced,
driving the equilibrium towards the products.
The temperature dependent, reaction kinetics, shown in Equation 3.1 (Smith, 1956).
^ - # § ) (3.1)
-2 .7 3 1 x 1 0 ^ k l = 0.083531 X 10 T-----
Reaction occurs in the hquid phase, in the reboiler, on the trays and in the condenser.
However, it is noted that due to the high boihng point of acetic acid it will be largely confined to the reboiler during the reaction phase and hence the bulk of reaction will occur in the reboiler. It is also important to note that the reverse reaction may weU occur on trays, particularly towards the bottom of the tray stack where large concentrations of water and ethyl acetate will occur. The heat of reaction is neghgible.
The calculation of the physical properties, notably the Vapour-Liquid equihbrium, is per formed using the Multiflash software (Infochem Ltd., 1998). Both the hquid and vapour phase are assumed to behave ideaUy; Raoult’s law apphes to the hquid phase and Dal ton’s law apphes to the vapour phase. The column configuration parameters and feed composition are shown in Table 3.1. The feed is charged to the reboiler, and some as holdup on trays and in the condenser. Rather than assume “perfect” pressure control in the condenser and level control in the reflux drum; PI controllers wiU be employed to con trol these quantities. In-order to avoid overshoot, particularly during composition control where the controller spends the initial period saturated as composition builds, anti-reset CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 55
windup features were added to the controller where integral error is not increased while
output is saturated.
As is normal industrial practice, the condenser pressure will be controlled by manipulation
of the condenser cooling rate. The level in the reflux drum, is controlled by manipulation
of the reflux flow, Lq. (Further discussion of appropriate pairings of controlled and manip
ulated variables is made in Chapter 5). The controller bias, controller gains and controller
reset times are given in Table 3.2. The condenser pressure is controlled at P = 0.5 bar.
Throughout the case studies, the reboiler heat duty is maintained at a constant value
Q = 0.885 X 10^ J/Sy except where it is considered as a source of disturbance in the
system. Two different operating policies are considered for producing a 0.6 mole fraction
ethyl acetate product, constant reflux ratio and controlled composition.
Constant reflux ratio operating policy
The operation using an open-loop, constant reflux ratio operating policy is considered.
For this operating policy, the column is charged and then operated at total reflux for 3
hours, then the internal reflux ratio (L/V) is fixed at 0.95 with product withdrawn into
the accumulator. This continues until the composition of ethyl acetate in the batch drops
to 60 mol%.
Controlled composition operating policy
The operation based on a feedback control strategy for both an undisturbed case and a
case with reboiler disturbances is also considered. The column is set up in exactly the same way as for the constant reflux ratio study except that a PI controller with anti-reset windup properties is introduced to control the distillate composition to 60 mol% ethyl acetate by manipulating the distillate flowrate, D. Controller parameters are shown in Table 3.2.
The batch is terminated when the distillate flowrate drops below, 1 x 10“^ m o l/s.
The controlled case study is also run with disturbances to the reboiler heat supply. The CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 56
Initial Material Holdups Column holdup 21.7 km ol condenser 100 mol trays 125 mol reboiler 20.6 km ol Feed Composition Ethanoic Acid 0.49 Ethanol 0.49 Ethyl Acetate 0 W ater 0.02 Tray Parameters Number of trays 8 Ytray 0.125 m^ hweir 10 m m ^wezr1 0.50 m A 0.1875 m^ Ah 0.025 m^ Reboiler Parameters Yyessel 0.5 m^ Heat Duty R885 xl05 J /s
Table 3.1: Column parameters
column is operated identically to the undisturbed case study but the reboiler heat duty is
stepped up and down, as shown in Figure 3.7.
3.2.2 Case study results
Figure 3.1 shows the values of the ethyl acetate composition in the distillate under constant
reflux and the flowrate of the distillate under these conditions. The simplified model
overestimates the distillate composition, particularly during the first 10 hours of the batch
which may explain why, when the composition tails off, it lags behind the rigorous. The
consequence of the overestimation is shown in in Figure 3.2 which shows the composition
profile for the accumulator where the composition of ethyl acetate in the accumulator remains higher for the entire batch. This is therefore why the batch terminates later for the simphfied model and the holdup of product is larger than for the rigorous model
(Table 3.3). CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 57
Condenser Pressure Control Controlled Var: Pc Manipulated Var: Q c Gain: K c -1 0 0 Reset: T£) 100 5 Set-Point: 0.5 atm Reflux Level Control Controlled Var: M t Manipulated Var: L Gain: K c -0 .1 Reset: T£> 100 5 Set-Point: 100 mol Distillate Composition Control Controlled Var: X d Manipulated Var: D Gain: K c - 1 Reset: tc 50 5 Set-Point: 0.6
Table 3.2: Controller parameters
The distillate flow (Figure 3.1) is, as expected for constant reflux profile, approximately constant after the initial total reflux period with a slight dip, towards the end of the batch as the reaction stops and the heavy product, water, begins to rise up the column.
Figure 3.3 shows the values of the composition of ethyl acetate with time with the values for the manipulated variable, distillate flowrate, for the two models for the controlled composition policy. As can be seen, the composition builds up as reaction proceeds and ethyl acetate is formed. During this period, the distillate flow is held at its minimum value, 0 i.e. no product is removed. As the composition passes through, 0.6, the distillate flowrate rises, starting at approximately, 1.5 hours., for the rigorous model. The distillate flowrate peaks after 5 hours then the flowrate dechnes. This distillate profile is quite different to that encountered for a non-reactive batch distillation separation (Figure 3.5) where the distillate flowrate rapidly increases from zero to its peak within a few minutes and then slowly decreases over the length of the batch. In the reactive case, there are two phenomena taking place simultaneously: the reaction which is the generation of ethyl CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 58
R igorous
0.6
0.4
Q 0.2
0 5 10 15 20 25 30 35
0.15
I E 0.05 i I 0
-0.05 0 5 10 1520 25 30 35 Time [Mrs]
Figure 3.1: Distillate composition (top) and distillate flowrate (bottom) for the constant reflux ratio study
acetate, principally in the reboiler, and the separation phenomena, extracting ethyl acetate
from the system. Figure 3.6 shows the reaction rate in the reboiler together with the
concentration of ethyl acetate. During the initial period, the reaction is dominant and the quantity of ethyl acetate in the column is low compared to the amount of ethanol (volatile
reactant). Hence the distillate rate will be low but will steadily increase as the quantity
of ethyl acetate increases in the column. The distillate rate reaches its maximum when the reaction begins to slow as the concentration of reactants in the still decreases and the
distillate flow rate decreases to maintain the composition of the distillate.
For the controlled policy there is a dramatic discrepancy between the simplified model and the rigorous model, again resulting from the simplified model over-estimating the composition of the column. This is reflected in the accumulator profile of composition and holdup, shown in Figure 3.4. The composition is held at 0.6 because of the controller but the amount of material in the accumulator builds up faster for the simphfied model. CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 59
1
0.8
; l 0.6
Io 0.4 — Rigorous Ü Simplified
0.2
0 0 5 10 15 20 25 30 35
14000
12000
10000
Ê 8000
2 6000
4000
2000
0 5 10 15 20 25 30 35 Time [hrs]
Figure 3.2: Accumulator composition (top) and holdup (bottom) for the constant reflux ratio study
This results in the, 20.7%, discrepancy in the batch times between the two modelling approaches (Table 3.3) with an insigniflcant 1.8% discrepancy between the predicted final accumulator holdups (Table 3.3).
In the third case study, the controlled reactive batch column is subjected to disturbances in the reboiler heat supply. The distillate composition profile, distillate flowrate and reboiler heat duty, as a percentage of the initial value, are shown in Figure 3.7. The profile for the distillate flow and composition are similar to the undisturbed case study. It is evident that there is a marginally different response in the distillate flows when the system responds to the reboiler disturbance. This may be due to the fact that the disturbances are being applied at different stages in the profile. For example, the first disturbance occurs while the distillate flow is still rising for the rigorous case study, but the simplified model has reach its peak distillate flow at this stage. Examining the final accumulator holdup. Table 3.3, there is an insiguificant discrepancy of 1.4% between the two modelling methods. However, CHAPTER 3. MODELLING OE RBD IN TRAY COLUMNS 60
0.7
0.6
0.5
E 0.4
0) 0.3 — R igorous
0.2
0 2 4 6 8 10 12
a 0.4
0.2
0 2 4 6 8 10 12 Time [hrs]
Figure 3.3: Distillate composition (top) and distillate flowrate (bottom) for controlled composition study
examining the predicted batch times (Table 3.3) the simplifled model simulation is 26.5%
shorter than the rigorous simulation. This is compared to the 20.7% discrepancy between
the batch times for the undisturbed case. It is apparent that the presence of disturbances
has resulted in a larger discrepancy between the two modelling approaches.
3.3 Conclusions
The objective of this chapter was to present a rigorous dynamic model for the simulation
of reactive batch distillation. A simphfied and more numerically robust model was also
presented for use in the control studies to be presented later (Chapter 5). In order to justify the adoption of the rigorous model over the simplifled model for simulations, the two modelling approaches were compared for the production of ethyl acetate. Two oper
ating policies: constant reflux ratio and controUed distillate composition were used for the CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 61
0.7 1 ' 1 1 1 1
0,6
- — Rigorous E 0,3 < " " Simplified Ü 0,2
0,1 : - 1 1 ] 1 _ i_ 0 10 12
12000
10000
8000
§- 6000
I 4000
2000
Time [hrs]
Figure 3.4; Accumulator composition (top) and holdup (bottom) for controlled composi tion study production of a 0.6 mole fraction mixture of ethyl acetate.
It appears that, although the predicted batch holdups are reasonably close, less than 5% discrepancy between the methods, the discrepancies in the predicted batch times are sig nificant and they increase as the column is operated under varying conditions. For the constant refiux ratio case study, the reflux fiow remains approximately constant through out. For the controlled case study, the refiux flow is changed continuously and for the disturbed case study, the reboiler disturbances introduce greater variances in the reflux fiow. This large difference in the performance of the simplified model compared to the rigorous under control would justify the adoption of the rigorous model for simulating reactive batch distillation. As reported by Sprensen and Skogestad (1994), reactive batch distillation cannot be effectively operated without some form of feedback control.
The simulation times for the two modelling approaches were markedly different. The simplified model took just 36 s to solve but the rigorous took 574 s, approximately 16 CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 62
ro.6
0.4
0.2
0 10 155 Time [hrs]
Figure 3.5: Distillate flow for non-reactive system
“ “ Reaction Rate [mol.m“ s” ] Ethyl Acetate Compostion [molfrac]
0.8
0.6
0 .4
0.2
0 2 4 6 8 10 12 Tim e [hrs]
Figure 3.6: Reboiler forward reaction rate and ethyl acetate composition (Rigorous Model)
times longer. The sixteen fold increase in computational time is particularly significant if the rigorous model were to be employed for optimisation. In dynamic optimisation, a complete simulation is required to evaluate the objective function at each step in the optimisation method. However, Monroy-Lopereba and Alvarez-Ramirez (2000) indicated that the controlled composition study can be thought of as being equivalent to the optimal reflux ratio policy. It is important to note that the “optimal” reflux ratio profile for the rigorous model is very different to that predicted for the simple model. It can therefore be assumed that optimisation results from the two models would also be significantly different. CHAPTER 3. MODELLING OF RBD IN TRAY COLUMNS 63
0.8
o 0.6
0.4 — R igorous 0.2 Simplified
0 2 4 6 8 10 12
E 0.5
110
= 105
® 100
cn 90
Tim e [hrs]
Figure 3.7: Distillate composition (top) and distillate flowrate (bottom) for controlled composition study with reboiler heat input disturbance (bottom)
Case Study Accumulator Holdup [kmol] Batch Time [hrs] Rigorous Simplifled % discr. Rigorous Simplified % discr. Const. Reflux 12.75 13.32 +4.4% 31.89 3R32 + + 4% Const. Comp. 10.46 10.65 +T 8% 10.57 8.39 -20.7% Const. Comp. (Disturb) 10.47 10.61 + L4% 10.64 7.84 -26.3%
Table 3.3: Accumulator holdup and batch times C hapter 4
Modelling of reactive batch distillation in packed columns
In this chapter, a dynamic rate based model is presented for the simulation
of reactive distillation in packed batch distillation columns. Packed columns
are often modelled using equilibrium stage models such as that presented in
Chapter 3. Here, a method is presented for determining the Height Equivalent to a Theoretical Plate (HETP) from the packed column model. The HETP is
demonstrated not to be constant, an assumption critical to equilibrium mod
elling. HETP is shown to be a strong function of both flowrate and compo
sition which vary with both time and position during batch operation. This
makes it difficult to determine a representative value for HETP. In this chap
ter, a packed column is compared to the rigorous and simplified tray columns
presented in Chapter 3, using an average HETP. The simulations are found to
be close where conditions vary slowly, e.g. under constant reflux ratio. How
ever, when conditions vary widely, for instance when the column is suffering
reboiler disturbances, the varying HETP and different dynamics combine to
produce large differences in performance. This demonstrates the need to adopt
a rate-based method instead of either the simplified or rigorous equilibrium tray
column models for modelling packed columns for control purposes.
64 CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 65
4.1 Rate-based modelling of packed columns
The objective of this chapter is to consider the dynamic modelling of packed columns, so that the control of this process can be examined. Therefore it is necessary to inves tigate the modelling alternatives: equihbrium or rate based, to estabhsh which is mores suitable. The equihbrium approach, presented in Chapter 3, has the advantage of signif icantly shorter simulation times but relies on the assumption that the packing behaves equivalently to an equihbrium tray column of a fixed number of trays. This approach assumes that the efficiency does not change significantly during the operation. The rate based method aims to model the packed column in a physically more reahstic manner than the equihbrium approach by considering the mass and energy transfer across the boundary between the hquid and vapour phases. The most important difference with this approach is the relaxation of the phase equihbrium assumption. In general, equihbrium is rarely attained in either heat or mass transfer as these are rate controUed processes driven by gradients in chemical potential and in temperature. In packed column modelhng, it is more appropriate to model the phases separately and to consider diffusional interaction phenomena between the phases described by the Maxwell-Stefan equations (Bird et al.
(1960) or Taylor and Krishna (1993)), as shown in Figure 4.1. The dynamic model of a catalytic packed column used in this thesis, is based on that presented in Furlonge (2000) which described a non-reactive packed distillation column. The model equations are given in Appendix A. In the following sections, the modelhng of the mass and energy transfer and the modelhng of the hquid and pressure dynamics in the packing, are discussed in more detail.
4.1.1 Modelling of mass and energy transfer
The determination of mass and energy transfer is carried out in accordance with the film theory, which assumes that hquid/vapour equihbrium exists only at the interface between the hquid and vapour phases (Welty et ah, 1984). Johnstone and Pigford (1942) concluded that, in distillation, the resistance to mass transfer is neghgible on the hquid CHAPTER 4 MODELLING OE RBD IN PACKED COLUMNS 66
LIQUID VAPOUR
REACTION
Figure 4.1: Mass and energy transfer between phases
side and therefore in the model used in this work, only the vapour side is considered in the calculation of the overall mass transfer coefficient. Heat transfer by conduction is described by heat transfer coefficients in both the licpiid and vapour films using the
Chilton-Colburn analogy (Welty et al., 1984). Heat is also transferred by convection as a result of the movement of mass from one phase to the other. Dispersion is not considered in this model but appropriate terms could be added to the mass and energy balances to account for this (Aly et ah, 1990a,b).
4.1.2 Modelling of hydrodynamics
In the absence of rigorous computational fluid dynamics, it is necessary to describe both the liquid phase holdup in tlie packing and tlie pressure drop relationships using correlations.
Ideally, one should develop these from the specific packing employed, using a suitable range of liquid and vapour flows (Kreul et ah, 1998). In this model, the liquid holdup- liquid flowrate relationship for Fall rings and Raschig rings is taken from Perry and Green
(1984). The pressure drop-vapour flowrate correlation is taken from Bemer and Kalis
(1978). CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 67
4.1.3 Modelling of chemical reactions
Typically, when considering reactive distillation columns, reaction is only considered in the hquid phase and can be homogeneously or heterogeneously catalysed. The modelhng of homogeneous catalysis is generahy simpler, requiring an appropriate kinetic expression.
There are a number of approaches to the modelhng of heterogeneously catalysed reactions in reactive columns, some of which are described below. In this chapter, the reaction is modelled as homogenous, with the reaction confined to the hquid phase. This permits heterogenous reactions to be modehed as quasi-homogenous if necessary.
Heterogeneous reaction (three phase model)
The three phase modelhng approach considers the sohd catalyst phase in addition to the hquid and vapour phases. It is useful as it ahows the characterisation of the various steps before and after reaction occurs. These are namely:
• Convective and diffusive transport of reactants through the hquid boundary layer to
the outer surface of the catalyst
• Internal diffusion through the catalyst structure
• Absorption of reactants on to the active catalytic sites
• Chemical reaction
• Desorption of the reaction products
• Diffusion through the catalyst structure to the surface
• Convective and diffusive transport through the hquid boundary layer
Experimental parameters are needed and their determination often difhcult. Hydrody namic conditions within the catalyst have to be determined using simphfied models such CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 68
as the Film Model and associated parameters such as the film thickness need to be de termined. It is also necessary to derive the appropriate sorption mechanisms for the determination of the rate constant.
Simplified heterogeneous reaction model
Some authors, for instance Thiel et al. (1996, 1997), have simplified the mechanism by assuming that the rate of reaction is dominated by one step in the sequence above. In their case, they assumed that the mass transport effects inside the catalyst particle play no role in relation to the kinetics of the chemical reaction. In other words, the mass transfer resistance of the chemical reaction wifi be very much bigger than that caused by mass transport. Another common assumption is to assume that it is the mass transfer step that is rate determining and the chemical reaction is assumed instantaneous.
Quasi-homogeneous reaction (two phase model)
The two-phase model is a simplification of the above models where the internal diffusion phenomena are neglected by incorporation into the rate constant. Kreul et al. (1998) pro vided justification for the use of this approach for certain reaction systems, particularly those where the sorption mechanism assumption breaks down. In some cases, particu larly for ion exchange resins where catalyst swelling occurs and the protons are evenly distributed throughout the resin, the system becomes very close to a homogeneous system and it is simpler to assume this rather than model the system as heterogeneous.
4.1.4 Packed column modelling
The column model is formed from the rate-based packing model for the column section, reboiler, condenser, refiux drum and accumulator models as given in appendix A. The packed column model consists of partial differential equations which are converted into differential and algebraic equations using an orthogonal collocation on finite elements CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 69
method (Wajge et ah, 1997) during integration in pPROMS (Process Systems Enterprise
Ltd., 1999).
4.2 Reactive batch distillation case study
In this section, the rate-based model is employed to simulate the production of ethyl acetate in a packed batch column. The case study wiU demonstrate how the separation efficiency of the packing is affected during the operation by calculating the HETP (Height Equivalent to a Theoretical Plate) at various positions in the packing. An equihbrium tray model rehes on the value of the HETP to be constant and it is demonstrated in this case study that this is not the case. It is therefore concluded that the equihbrium approach is not suitable for modelhng packed batch columns for control purposes.
4.2.1 Column design
In order to demonstrate the behaviour of reactive batch distiUation in the packed column the ethyl acetate case study presented in Chapter 3 wiU be used. As in Chapter 3, the more rigorous, temperature dependent kinetics presented by Smith (1956) are employed and ideal physical properties are adopted, modelled by Multiflash (Infochem Ltd., 1998).
The synthesis is performed in a regular packed column flUed with 25mm steel Pah rings.
Table 4.1 details the column dimensions and the packing features. The feed is charged to the reboiler and some as holdup on the trays and condenser. In this model, perfect pressure control is assumed. However, level control in the reflux drum, necessary to prevent the drum from overflowing or draining, is performed by a PI controUer manipulating the reflux flow, Lq. The controUer bias, controUer gain and controUer reset time are as given in Table 3.2 in Chapter 3. The condenser pressure is fixed at P = 0.5 bar. Throughout the case study, the reboiler heat duty is maintained at a constant value, Q = 0.885 X 10^ J /s , except where it is considered as a source of disturbance in the system. CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 70
Accumulator Volume 0.4 rmP Reflux drum Volume 0.01 rrN Packed bed Total length 8 m Diameter 0.4 m Packing PaU Rings Reboiler drum Cross-sectional area 4 m? Volume 0.65 wP Initial Charge 21.7 km ol Ethanoic Acid 49% Ethanol 49% W ater 2% Ethyl A cetate 0% PaU Ring Size 25 m m Characteristics Material Steel (Coulson and Critical Surface Tension 75 N /m Richardson, 1991) Packing Area {Sb ) 210 m ^/m ^ Packing Factor {F) 160 m~^ Void Fraction e 0.94
Table 4.1: Column dimensions and packing characteristics
4.2.2 Column operation
The column is operated under two different operating scenarios, constant reflux ratio and controlled composition operating poUcy.
Constant reflux ratio operating policy
In part of this study, the column is operated using an open-loop, constant reflux ratio operating policy. For this operating pohcy, the column is charged and then operated at total reflux for 3 hours, then the internal reflux ratio {L/V) is fixed at 0.95 with product withdrawn into the accumulator. This continues until the composition of ethyl acetate in the batch drops to 60 mol% (after approxim ately 30 hrs). CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 71
Controlled composition operating policy
The operation using a feedback control strategy for both an undisturbed case and a case with reboiler disturbances is also considered. The column is set up in exactly the same way as for the constant reflux ratio study except that a PI controUer with anti-reset windup properties is introduced to control the distiUate composition to 60 mol% ethyl acetate by manipulating the distiUate flowrate, D. The batch is terminated when the distiUate flowrate becomes negligible, below 1 x 10“^m o lls.
The controUed case study is also operated with disturbances in the reboiler heat supply.
The column is operated identically to the undisturbed case study but the reboiler heat duty is stepped up and down (shown in Figure 4.10). The principle objective of using the same case study as that used for the tray columns is so that the two modelling approaches can be compared. Two important issues are investigated. Firstly, it is necessary to estabUsh the degree of discretisation acceptable for the simulations, i.e. the number of finite elements for orthogonal coUocation to be employed. Secondly, so that the packed column model can be compared to the rigorous and simplified tray column models, presented in Chapter 3 an equivalent number of trays must be estabUshed. This is done by analysing the packed column model and establishing a suitable value for the HETP.
4.2.3 Effect of discretisation
As mentioned earher. Orthogonal Collocation on Finite Elements Method (OCFEM) is used to discretise the axial dimension within the column packing. In this section, the effect of the number of finite elements used during the simulation results is investigated.
The greater the number of finite elements adopted, the more accurate the simulation but at the expense of greatly increased computational time. It is therefore prudent to adopt the smallest number of finite elements possible without compromising accuracy. How ever, adopting a greater number of finite elements also gives more detailed information about how quantities vary within the packing and when analysing HETP, it is necessary to have this detailed information, hence a larger number of finite elements may be appro CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 72
priate. In Table 4.2 are displayed the batch times and accumulated product for the two
operating policies (constant reflux ratio and controlled composition) modelled with third
order orthogonal coUocation on two and on four flnite elements. For the constant reflux
ratio poUcy the discrepancy between the predicted batch time and holdup of product in
the accumulator is neghgible. However, for the controUed composition poUcy, there is a
1.27% discrepancy between the predicted batch times with neghgible difference between
the predicted accumulator holdup. Due to the changing flowrate and therefore greater
dynamic change, a higher degree of discretisation may be needed. It is important to note
that the 1.27% discrepancy in batch time may not justify employing the higher degree of
discretisation due to the vastly increased computational times, 765 s to 4113 5, also shown
in Table 4.2.
Therefore, third order orthogonal coUocation on two finite elements wiU be adopted for
the simulations in this thesis. However, this level of discretisation yields only seven points
across the discrete space so four flnite elements where the number of points increases to thirteen, wiU be used for the HETP investigation.
2 Finite Elements 4 Finite Elements % Discrepancy Constant Reflux Ratio: Batch Time [hr] 32.653 32.625 -0.09% Product Holdup [kmol] 12.824 12.835 -0.09% Simulation Time [s] 1523 7592 +398% ControUed Composition: Batch Time [hr] 10.457 10.326 -1.27% Product Holdup [kmol] 10.296 10.283 -0.13% Simulation Time [s] 765 4113 +438%
Table 4.2: Comparison between level of discretisation for 8m packed column
4.2.4 Determination of HETP
As already mentioned, HETP, the Height Equivalent to a Theoretical Plate, is a method employed to reconcile column packing with equilibrium stage modeUing. Authors (Treybal
(1980) and Kreul et al. (1999)) have already concluded that HETP varies, not only with CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 73
type and size of packing, but also strongly with howrates of both liquid and vapour phases
and by component concentration. Thus, one would naturally expect a spatial variation
in a continuous column operating under steady state. In a batch column, one would in
addition also expect a strong variance with time. The difficulty with this approach hes
in the failure to account for, in a physically reahstic way, the behaviour of packing by
assuming a staged equihbrium model. In this section, it is considered how HETP varies
within the packing section and with time. The motivation is two fold, firstly to confirm
that HETP variance is too great to justify equihbrium modelhng and secondly to examine
how changing conditions within the column affect the performance of the packing.
Calculation of HETP
In order to estimate the number of trays equivalent to that of the packed column it is
necessary to assess how the “Height Equivalent to a Theoretical Plate” (HETP) varies within the packing. For this purpose, a column packing with a height of 8m was selected
and axially discretised using third order orthogonal coUocation on four finite elements,
this level of discretisation yields 13 discretisation points within the body of the packing.
Figure 4.2 shows the composition of ethyl acetate in the vapour phase as a function of
height together with the composition of the vapour at thermodynamic equilibrium with the
liquid phase for the constant reflux ratio policy after 3 hrs of operation. The equihbrium
composition at the bottom of the column is approximately 0.45. This is equivalent to the
vapour outflow composition of the first theoretical plate. This value is achieved on the
vapour composition curve at approximately 1 m. In other words, the first 1 m of packing
achieves the equivalent separation of one equihbrium stage. Therefore, the HETP of the
packing at the bottom of the column is 1 m. The HETP at any given location is simply the horizontal distance between these curves (Furlonge, 2000).
The HETP was analysed for the constant reflux ratio pohcy and for the controlled com
position case study. Figure 4.3 shows the profiles of composition of ethyl acetate in the CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 74
0.85
0.8
0.75
0.7
o 0.65
0.5
0.45 — Actual - - Equilibrium 0.4 HETP
0.35
Height [m]
Figure 4.2: Vapour composition of Ethyl Acetate at 3 hrs (constant reflux ratio)
reboiler and distillate and the distillate flow during the batch. The top two plots show the
profiles for the constant composition policy and the bottom two plots show the profiles
for the controlled composition policy. It is evident that the composition varies at the top
and the bottom of the column for the constant reflux ratio policy, albeit slowly for the
majority of the batch due to the high reflux ratio. There is a rapid drop in the distillate
composition after 25 hours. Here, the heavy components, predominantly water, begin
to be removed. This rapid drop in distillate composition accompanies a shght drop in
distillate flow, which otherwise remains approximately constant at 0.12 m o l/s.
For the controlled composition pohcy, composition at the top of the column remains
constant past the total reflux period due to the controller. The reboiler composition
increases to a maximum of about 0.2 at 3 hours and then decreases over the remainder of the batch. The most dramatic variation in the column is that of the distiUate flow, manipulated to control the composition. The distillate increases to a peak at about 5 CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 75
hours during the main reaction phase and then decreases towards the end of the batch
as the rate of production of ethyl acetate begins to tail off. The consequence of this is
that the flow of liquid reflux back into the packing is initially high, decreases during the
middle portion of the batch and then increases again towards the end of the batch. This
is thought to have a profound effect on the efficiency of the column and consequently on
the HETP.
The actual and equilibrium vapour compositions (as shown in Figure 4.2) were extracted
at a number of time steps throughout the batch process. A program was written within
Matlab (The Math Works Inc., 1996) to interpolate the data using Lagrangian polyno
mials and to calculate the horizontal distance between the curves as a function of height
(Equivalent to HETP). It should be noted that since the actual composition curve does not extend past the top of the column it is not possible to calculate past the HETP at the top of the column.
For the constant reflux ratio policy a series of axial HETP profiles for different times during
the batch are shown in Figure 4.4. The HETP is approximately constant throughout the
packing between 0.78m and 0.91m for the first three time steps (6 hrs to 18 hrs). There
is a shght drop in HETP at the bottom of the column which becomes greater as time
progresses. At 24hrs, this becomes most significant with the HETP dropping to 0.62m at
the bottom and rising with height in the column. From Figure 4.3 it was shown that the
flowrates within the column remain approximately constant and therefore the variation in
HETP is likely to be due principally to the change in concentration within the packing
during the batch. The mean HETP against time is shown in Figure 4.5. The mean HETP
starts at about 0.9 m at the start of the batch and decreases slightly to approximately
0.86 m for most of the batch up to 21 hrs where it dips shghtly down to 0.8 m at 21 hrs.
Generally, the HETP falls as the batch proceeds which means the packing becomes more efficient towards the end of the batch. The average HETP with time and space for this policy is 0.857 m
For the controlled composition case study a series of axial HETP profiles for different times during the batch are shown in Figure 4.6. Up until the peak in distillate flow at 5 CHAPTER 4. MODELLING DE RBD IN PACKED COLUMNS 76
hours, the HETP is approximately constant throughout the packing. As the reflux flow falls, the HETP falls. After the peak at 5 hours, greater variation in the packing occurs with the bottom and top portions of the column decreasing in efficiency and the central section between 2 m and 3 m continuing to increase in efficiency. The mean HETP against time is shown in Figure 4.7. The mean HETP appears to approximately follow the change in distillate flowrate, shown in Figure 4.3. The HETP decreases from its maximum of
0.97; m, and hence efficiency increases as the flowrate begins to rise to its maximum at 5 hours. The HETP reaches its minimum of 0.725 m at 6 hours and rises again towards the end of the batch. This indicates that HETP is a strong function of flowrate and increases with increasing liquid reflux. The average HETP with time and space is 0.825 m. CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 77
Distillate Reboiler ^ 0.8
0.6
0.4
E 0.2
20 30 35
0.14
0.12
S 0.08
« 0.06
S 0.04
0.02
0 5 10 15 20 25 30 35 Time [hrs] 0.8 Distillate Reboiler 0.6
B 0.2
0 0 1 2 3 4 5 6 7 8 9 10
0.7
0.6 o 0.5
5 0.4
® 0.3
0.2
0 1 2 3 4 5 6 7 8 9 10 Time [hrs]
Figure 4.3: Composition and Distillate Flowrate - (TOP Constant reflux ratio - BOTTOM Controlled Composition) CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 78
0.9 5
0.9
0.8 5
0.8 — 6 hrs
18 hrs ^ 0.75 24 hrs
0.7
0.65
0.6
Height [m]
Figure 4.4: HETP packing profile against time for Reactive Case Study (Constant reflux ratio)
0.92
0.9
0.86
® 0.84
0.82
0.8 20 25 30 Time [hr]
Figure 4.5: Mean HETP profile for Reactive Case Study (Constant Reflux Ratio) CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 79
1 hr - - 3 hr
£ CL tD X
0.9
0.8
0.7
0.6
height [m]
Figure 4.6: HETP packing profile against time for Reactive Case Study (Controlled)
0.95
5 0.9
0 .85
0.75
0.7 1 2 3 4 56 7 8 9 Time [hr]
Figure 4.7: Mean HETP profile for Reactive Case Study (Controlled) CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 80
4.3 Comparison between rate based and equilibrium models
The most important conclusion from the study in the previous section is that HETP is
not constant within the column section or with time. This has potentially severe imphca-
tions when adopting the equihbrium modelling approach to model packed batch columns,
especially when hquid flowrate is varying during the batch such as during the controlled
composition policy. A rate-based approach is therefore expected to provide a more accu
rate representation of a packed column. To confirm this, the two approaches are compared.
In order to make this comparison it is necessary to ascertain the number of equilibrium trays that corresponds to the 8 m of packing. This is difficult to do due to the variation
experienced in both the constant reflux ratio and especially in the controlled composition
policy. Using the average HETPs of the two operating policies, 0.857 m for the constant reflux ratio and 0.825 m for the controlled composition gives columns of 9.3 trays and
9.7 trays respectively which both round up to 10 trays. It is also important, due to the
high variations in HETP, to investigate the impfications of selecting values of HETP at
the extremes. The HETP varies most significantly for the controlled composition policy:
between 1.4 m and 0.6 m corresponding to a 6 tray and 14 tray column respectively.
Therefore, in this section the 8 m packed column is compared to the rigorous tray col
umn model (Chapter 3) with 6, 10 and 14 trays. The packed column is also compared to
the simplified tray model with 10 trays, to investigate the implications of this approach
to packed column modelling. Comparisons are made under the three different operating
policies: Constant reflux ratio [R = 0.95), constant composition of 0.6 mole fraction Ethyl
Acetate and the constant composition policy under disturbances.
In Table 4.3 are shown the batch times and total product accumulated with composition
0.6 mole fraction ethyl acetate. The table also shows the percentage discrepancy between
the different models as compared to the 8 m rate based packed column model.
The constant reflux ratio, distillate composition and flow profiles are shown in Figure 4.8.
The performance of the rigorous and simplified tray column configurations are close to that of the packed column model. The 10 tray rigorous model, derived from average CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 81
HETP corresponds most closely of the tray column models. It is noted that the simplified
10 tray model responds much quicker and has a similar initial response to the 14 tray
column. The distillate profiles are also close with the distillate flow marginally lower for the
packed column model. The close profiles are reflected in the batch times and accumulator
holdup values (Table 4.3). There is no discrepancy between the packed column model
and the rigorous 10 tray model for batch time with a slight overestimate of 2.0% for the
accumulator holdup. For the simplified 10 tray model, there is an overestimate of 1.9%
in the batch time and an overestimate of 3.5% in the accumulator holdup. The smaller 6
tray rigorous column underestimates both the batch time, —5.1% and holdup, —3.5% due
to its poorer separation performance. Similarly, due to its higher separation ability the
larger 14 tray rigorous column overestimates the batch time, +3.6%, and holdup, +5.9%.
The controlled composition profiles are shown in Figure 4.9. As with the constant reflux
ratio profiles the profiles of the packed column are close to those of the rigorous tray
columns. However it is interesting to note that, due to the controllers, there is oscillation
in the distillate flowrate in the region of 2 hours. The amplitude of the oscillation is larger
for the packed column than for of the rigorous tray columns except that of the largest 14
tray rigorous tray column. This suggest a slightly slower dynamic response of the packed
column and the larger tray column. This slower response is also reflected in the distillate
composition where the packed column and the larger tray column take slightly longer to
reach the controller set point. In terms of final product accumulated and batch times
the variance between the models is larger than for the constant composition case study
(Table 4.3). The simplified 10 tray model has very significant 21.1% shorter batch time
and 2.8% larger product accumulated. The rigorous 10 tray model is still close to the
packed column model, —0.8% batch time and +1.2% holdup . The larger and smaller rigorous tray columns are more significantly different than for the constant reflux ratio policy. The discrepancies for the 14 tray column are 4.2% shorter batch time and 0.6% larger holdup. For the smallest, 6 tray column the batch time is 15.3% longer and the holdup is 2.0% larger.
The controlled composition with reboiler disturbance profiles are shown in Figure 4.10. CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 82
Due to the reboiler disturbances there is a larger difference between the packed column profiles and the rigorous tray column profiles. However, the packed column profiles remain much closer to those of the rigorous tray column models than to those of the simpbhed tray column model. The reboiler disturbances produce oscillations in the distillate flow which is most pronounced for the packed column and the largest, 14 tray rigorous column.
The oscillations become more pronounced towards the end of the batch. The discrepancy in accumulator holdup is small at about (2%) for both columns. The discrepancy in batch times and accumulator holdups (Table 4.3) are generally larger than for the controlled policy without disturbances. The variance is greatest for the batch times. The simplified
10 tray model underestimates the batch time by 20.1%, the rigorous 10 tray model overes timates by 3.2%. The larger, 14 tray column underestimates by 2.1%. The smaller, 6 tray column overestimates by a significant 25.8%. In terms of accumulator holdup aU four tray column models overestimate the amount of holdup in the accumulator. The simplified 10 tray model overestimates by 3.0%, the rigorous 6 tray by 2.6%, the rigorous 10 tray by 1.6% and the rigorous 14 tray by 0.9%. CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 83
Model Size Batch Time Acc. Holdup Hours % discrepancy Holdup % discrepancy Constant Reflux Packed (NEQ) 8 m 32.65 12.82 Simplified (EQ) 10 Trays 33.27 +1.9 % 13.27 +3.5 % Rigorous (EQ) 6 Trays 30.99 -5.1 12.37 -3.5% Rigorous (EQ) 10 Trays 32.65 0 % 13.08 +2.0 % Rigorous (EQ) 14 Trays 33.84 +3.6 % 13.58 +5.9 % Controlled Packed (NEQ) 8 m 10.15 10.30 Simplified (EQ) 10 Trays 8.01 -21.1 % 10.59 +2.8 % Rigorous (EQ) 6 Trays 11.70 4-15.3 %o 10.51 +2.0 % Rigorous (EQ) 10 Trays 10.07 -0.8 %o 10.42 +1.2 % Rigorous (EQ) 14 Trays R72 -4.2 % 10.36 +0.6 % Controlled + Disturbances Packed (NEQ) 8 m 9.34 10.24 Simplified (EQ) 10 Trays 7M6 -20.1 % 10.55 +3.0 % Rigorous (EQ) 6 Trays 11.75 +25.8 % 10.51 +2.6 % Rigorous (EQ) 10 Trays 9.64 +3.2 % 10.40 +1.6 % Rigorous (EQ) 14 Trays 9.14 -2.1 10.33 +0.9 %
Table 4.3: Comparison between modeUing approaches, (EQ: equilibrium model, NEQ: rate based model) CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 84
Packed (8m)
■ Rig 10 Tray ' Rig 14 Tray ■ Simpl. 10 Tray
0.4
Q 0.2
0 5 10 15 20 2530 35
0.15 f II 0.05 I
-0.05 0 5 10 15 20 25 30 35 Time [hrs]
Figure 4.8: Distillate composition (top) and distillate flowrate (bottom) for constant reflux ratio policy
0.7 " Packed (8m) ' - ' Rig 6 Tray - - Rig 10 Tray ■sO.5 Rig 14 Tray Q- Simpi. 10 Tray E 0.4 Ô S 03 1 0.2 b 0.1
0 10 12
0.6
0.4
Q 0.2
Time [hrs]
Figure 4.9: Distillate composition (top) and distillate flowrate (bottom) for controlled composition policy CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 85
c 0.8 wmm Packed (8m) ^06 Rig 6 Tray - - Rig 10 Tray I 0.4 Rig 14 Tray — Simpl. 10 Tray .= 0.2
10 12
iZ 0.5
Time [hrs] CFIIO
S 105 6 Time [hrs] Figure 4.10: Distillate composition (top), distillate flowrate (middle) and reboiler distur bance profile (bottom) for controlled composition policy CHAPTER 4. MODELLING OF RBD IN PACKED COLUMNS 86 4.4 Conclusion In this chapter, the production of ethyl acetate in a packed reactive batch distillation column has been considered. Packed columns can either be modelled as an equivalent tray column by determining a suitable value for the HETP (Height Equivalent to a Theoretical Plate) or more physically reahstically by a rate-based model. The objective of this chapter was to establish whether or not, the less physically realistic, tray column model would be suitable for modelling the packed column for reactive purposes. A packed column model which extended the work of Furlonge (2000) to include chemical reaction throughout the liquid phase was presented. This model was used to simulate the behaviour of the constant reflux ratio and controlled composition case study presented in the previous chapter, in a packed batch column. A method was presented for establishing the HETP for the column and this was used on both case studies. It was concluded that the HETP is not constant, varying with time and packing height as a result of varying composition and liquid flowrate. Consequently, the controlled composition case study was found to vary more than the constant reflux ratio case study due to greater changes in liquid flowrate. A comparison was made between the packed column model and rigorous and simplified tray column models with 10 trays, which corresponded to the average HETP found for both the constant reflux ratio policy and the controlled composition policy. The packed column was also compared to rigorous tray columns with 14 and 6 trays which are the columns corresponding to the maximum and minimum HETP values in the controlled composition study. The rigorous column model with 10 trays agreed most closely with the packed column model while the simplified model and the larger and smaller rigorous column models had poorer agreement. This suggest that if the equilibrium approach is taken then it is important to use the average HETP and to model the column using the rigorous tray column model. However, for the controlled case study with and without disturbances the constant HETP assumption begins to break down due to the higher variance in HETP experienced within the packing due to the varying flowrates. The CHAPTER 4. MODELLING DE RBD IN PACKED COLUMNS 87 greater variance may also be due to the different dynamics within the columns leading to the slower behaviour of the packed column. The variation in HETP was found to be too great to justify the adoption of the equilibrium approach for modelhng packed columns especially where liquid flowrates vary considerably, such as during control. Therefore, in order to study control and controllability of packed columns it is necessary to adopt a modelling approach that accounts for the physically different mechanisms of heat and mass transfer encountered in a packed column and the rate-based packed column model will be adopted when investigating packed column control in Chapter 5. C hapter 5 Control of reactive batch columns In this chapter, the control of reactive hatch distillation in both tray and packed columns is considered. It is important to have an appreciation of how design decisions affect column control as this insight is incorporated into column de sign and operation. Optimal design and operation may not be achievable with out considering control simultaneously. Both a simulation based approach and a frequency response approach to controllability is considered. The simula tion approach requires a control scheme to be implemented and is more time consuming. The frequency response approach requires linear approximations to non-linear process models. A number of linear models are needed to track changes in controllability through the batch. A robust method of linearisation is presented to deal with numerical problems arising from the rigorous tray col umn model. These methods are applied to tray and packed columns and are used to identify how the choice of column and column size affects controllabil ity. The effects of batch time and reaction are also investigated. It is concluded that packed columns are harder to control than tray columns, control becomes more difficult for larger columns, particularly for packed. For this case study, control is found to get harder towards the end of the batch and reaction is found to have negligible direct effect on control. However, reaction affects the reflux ratio profile which is found to have a significant effect on controllability. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 89 Disturbances (d) Controlled Outputs (y) Process Measured Outputs ( y j Inputs (u) Figure 5.1: Process for control 5.1 Introduction A general process that is to be controlled is illustrated in Figure 5.1. Initially, the require ments of a control system need to be developed. These control objectives are formulated such that the operational requirements for the process can be achieved. The process’ oper ational requirements may be to perform optimally, maximising profit or minimising batch time, for instance, with reference to safety, environmental regulations, product specifica tions and operational constraints. It is important to consider how easily these control objectives can be met for the given process. Skogestad and Postlethwaite (1996) defined input-output controllability as being this ability to meet the control objectives. They formally defined it as: Input-output controllability is the ability to achieve acceptable control perfor mance; that is, to keep the outputs (y) within specified bounds or displace ments from their references (r), in spite of unknown but bounded variations, such as disturbances (d) and plant changes, using available inputs (u) and available measurements (y ^ and dm)- With this definition of controllability in mind, it is also important to consider which measurements (y-m, dm) should be made and which inputs (u) should be manipulated to achieve the best possible performance. Seborg et al. (1989) presented a series of guidelines for doing so: CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 90 1. Select control outputs that are not self-regulating 2. Select control outputs that have favourable dynamic and static characteristics, i,e. there should exist an input with a significant, direct and rapid effect 3. Select inputs that have large effects on the outputs 4. Select inputs that rapidly affect the controlled variables It is important to stress that controllability is an inherent property of a process and can only be affected by physical changes to the process. Therefore, it is important to understand how any modifications made to the plant affects its controllabihty. There are a number of approaches to a controllability analysis: The most common ap proach is the simulation approach which involves a candidate control system being im plemented and then tested against a range of disturbances and set point changes whilst evaluating the performance. Alternatively, a frequency response approach can be taken to assess the controllability of a process. In order to use the frequency response approach, it is necessary to use linear approximations of the non-linear process models. For continuous processes, this would typically be linearisations about the steady-state. However, for batch processes where the operating point changes with time, it is necessary to bnearise at multiple points throughout the batch. Rather than representing the deviation away from steady-state, the linear models represent the deviation away from the operating trajectory. The frequency response approach has the advantage that it can be applied to open-loop processes, thereby eliminating the effect of the controller on the controllability analysis. A simulation approach and a frequency response approach to controllabihty analysis will be adopted in this chapter. Firstly, the methods for controllabihty analysis wiU be outlined for both the time domain and the frequency domain. Then a method for applying the controllabihty tools will be outhned, together with a method for deaUng with numerical problems in the rigorous tray column model. These methods are then used to investigate CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 91 Condenser Reflux Drum D istillate DReflux L t Distillate DReflux Vapour Boilup Reboiler Figure 5.2: Batch Column the controllability of the tray column and the packed column presented in Chapters 3 and 4. 5.1.1 Control of batch distillation columns In both non-reactive and reactive batch distillation, shown in Figure 5.2, there are a number of common control objectives: • Condenser Pressure Control: Typically controlled through the manipulation of con denser duty • Reflux Drum Level Control: Normally controlled through the manipulation of reflux or distillate flow • Product/ distillate composition control: Normally controlled through the manipula tion of reflux or distillate flow Speciflcally for some reactive batch distillation operations: CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 92 • Reboiler temperature control Condenser pressure control Condenser pressure control is a requirement of all distillation columns. This is typically achieved through the manipulation of the condenser duty. Pressure dynamics are fast compared to other distillation phenomena which is the justification for assuming perfect pressure control taken by some authors as discussed in Chapter 3. Reflux drum level control Reflux drum level control is necessary in all types of columns as this is not a self regulating process. It is necessary to maintain a “reasonable quantity” of inventory in the reflux drum to help dampen disturbances in the vapour flowrate entering the condenser. Normally either hquid distillate flowrate, D or the reflux flowrate, Lt-, is used for this purpose. Distillate composition control In non-reactive batch distillation, a number of open-loop pohcies can be employed to achieve the desired product composition in the accumulator. This would entail either a fixed reflux ratio or time varying reflux ratio, both implemented through the manipulation of whichever out of the reflux flowrate, Lt or distillate flowrate, D is not being used for level control. Sprensen and Skogestad (1994) concluded that, for reactive batch distillation, it is not rehable to apply these open loop pohcies and that some form of feedback control is required. Here the composition of the distiUate, is maintained at a desired value through the manipulation of either the reflux or distiUate flowrates. Reaction temperature control Sprensen and Skogestad (1994) indicated that for some reactions it is necessary to control the temperature of the reboiler, where reaction tends to occur, to within tight bounds to CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 93 ensure a consistent product between batches. Temperature can be controlled by manipu lating the reboiler boilup rate hg. Two candidate control schemes (Table 5.1) for reactive batch distillation are considered by Sprensen and Skogestad (1994) although they considered perfect level control in the reflux drum. Skogestad (1997) indicated that for continuous distillation, there is greater interaction with the level control loop when employing Scheme 1 (DV) than with Scheme 2 (LV). M x XD T b P c Scheme 1 Lt D Qc Scheme 2 D Lt Vb Qc Table 5.1: Control Schemes Equipment modifications As indicated earlier, controllability is a property of the process only and is independent of the control system. It can only be affected by modification of the process equipment. Some modifications that can be made to the process to modify controllability are outhned below. • Modifying the size of the reflux drum or adding extra buffer tanks wiU dampen disturbances. • Changing the height of the column or type of column will affect controllabihty. Larger distances between top and bottom of the column has a decoupling effect on control loops at the bottom and top. • The choice of measuring equipment can introduce significant delays and therefore control difficulties into the system. Sneesby et al. (1997) discussed composition mea surement and reported that these fah into three categories: directly, using one or more online analysers; indirectly, using temperature to infer composition; and exter nally, by taking samples and measuring in the laboratory. Using analysers has many CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 94 advantages but is expensive, requires regular maintenance, and often introduces sig nificant time delay into the process. Inferential control is cheaper but can be less accurate. External measurement is useful for monitoring the process but cannot be used for closed loop control. Sneesby et al. (1997) also noted that inferential control needs to be used with caution and placements need to be made where the change in temperature, accurately reflects changes in composition. 5.2 Controllability methods In this section, controllability methods for both the time domain and the frequency domain as presented in the literature are reported. 5.2.1 Simulation controllability analysis The simulation or time domain approach requires that a candidate control system for the process control system is designed and implemented. The controllers would be tuned by some method such as that presented by Ziegler and Nichols (1942). The performance of the selected control system can then be evaluated against a set of likely disturbances and set-point changes. The most controllable configuration would be the one with the “best” performance. Which criteria are adopted for evaluating this “best” performance will depend on the apphcation. The tuning method described by Ziegler and Nichols (1942) was designed to give good overall performance. However, some applications, notably level control, require that the disturbances to the manipulated flow are kept to a minimum rather than tight level control, so that they are not passed on downstream. An essential requirement of aU control systems is the avoidance of both input and output constraints. These constraints are requirements of both safety and quality. In the case of manipulated variables, the wear and tear of the valves must be considered and rapid movement of the valves between fully open and fully shut should be avoided. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 95 Time domain performance indicators that can be applied to both the controlled output {y) and the manipulated variables (u) are reported by Skogestad and Postlethwaite (1996). Those that relate to the speed of response: • Rise time: Time taken to reach 90% of its final value should be small. • Settling time: Time after which output remains within ±5% of its final value. Nor mally required to be small. Those that relate to the quality of response: • Overshoot: Peak value divided by the final value should be less than 20%. • Decay Ratio: Ratio of second and first peaks, which should be less than 0.3. • Steady-State offset: Difference between the final value achieved and the desired value. This is often required to be small. Shinskey (1994) mentioned a number of performance indicators based on the integration of the controller error. As well as indicators of performance, controller settings can be optimised through minimisation of these functions. Integrated error (IE) pt2 1E = edt (5.1) Jti The error can be either positive or negative and therefore an Integrated Error of zero could be obtained with a continuously oscillating system. Consequently, this is not a measure of stability. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 96 Integrated absolute error (lAE) rt2>t2 lA E = / |e| dt (5.2) Jti This represents the total area under the response curve on both sides of zero error. It is an improvement over Integrated Error as it incorporates stability. The lA E will only tend to a finite value for a stable loop which achieves its set point. Integrated square error (ISE) ISE = / e^dt (5.3) Jti This is similar in nature to the Integrated Absolute Error from a stability point of view since the squaring of the error removes the negative sign. However, also due to the squaring, larger errors gain a greater weight from smaller errors. Shinskey (1994) indicated that, as a consequence of this, when optimising controller parameters by minimising the ISE, longer settling times are encountered than for minimising lAE. Integral of time and absolute error (ITAE) ft2 I T A E = / t\e\dt (5.4) Jt2 This measure puts greater weight on the setthng time and as such has an opposite effect on the optimal parameters than ISE, tending to penalise long term errors more than short term errors. The minimum ITAE response wiU tend to have higher peak response and shorter settling time than minimum lAE and ISE responses. Shortening the settling time is particularly useful in batch processes where it is an economical advantage to achieve changes in states as quickly as possible when set-point changes are made. Sum m ary The drawback with the simulation approach, is that it is impossible to guarantee that the conclusions drawn are due to the inherent properties of the process, which is of primary CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 97 interest. The control system selected, or for that matter, the set of disturbances and setpoint changes chosen, may well be responsible. 5.2.2 Frequency response controllability analysis In order to overcome the limitations of the simulation approach, Skogestad and Postleth waite (1996) indicated that it may be more useful to assess controllability in the frequency domain rather than the time domain. The frequency domain approach uses Frequency Analysis of the process. From this, information about the process’ controllability can be derived. This technique can be applied to an open loop process without controllers, and can also be usefully applied to open loop processes with controller for controller design and also in closed loop processes. As mentioned earlier, one cannot use the simulation approach on uncontrolled, unstable processes. Thus the frequency domain approach offers the advantage of being controller independent, the main shortcoming of the simulation approach. A brief overview of the nature of frequency analysis will be presented, followed by a description of controUabiHty measures that can typically be appbed. Frequency analysis The principle behind frequency analysis is that a sinusoidal input is appbed to a process, after a long time, the output will oscibate with exactly the same frequency as the input signal. However, the output wib be out of phase with the input [phase lag) and the ampbtude wib change [amplitude ratio). As the frequency of the impinging sinusoid changes so wib the values of the phase lag and amplitude ratio. These are ibustrated for a first order process, e.g. buffer tank, in Figure 5.3 where as frequency increases the ampbtude of the output signal decreases and phase shifts to being 90° behind the input signal. Controbabibty results about the process can be interpreted from this information. It should be noted that although a sinusoidal signal is not a typical process input, Fourier analysis shows that any discrete signal is a compound of sinusoids at different frequencies. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 98 §5 ° 2 ,-2 0 M -60 É -80 1010 10 10' Frequency (rad/sec) Figure 5.3: Frequency response of a first order system Controllability in the frequency domain In order to use frequency response controllability tools, it is necessary to use linearised versions of the non-linear process models with the assumption that the hnear model ac curately describes the non-hnear model for a brief period after linearisation. It is vital to confirm that this is indeed the case through comparison with non-hnear simulation. It should be noted that, for highly non-hnear processes such as distiUation columns, the hnear models wih not accurately describe the process state at large deviations. When operating in batch mode, where the states are continuously changing, it is necessary to hnearise the process model at multiple points to appreciate how controhabihty changes during the process in order to design the controllers. Thus in order to carry out a controhabihty analysis of a plant one requires a sufficiently accurate, normaUy non-hnear, model describing the operation of the plant. A general procedure for this would be: • Simulate non-hnear plant model to desired point of hnearisation. • Generate hnearised approximation to model Apply controhabihty measures CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 99 e=r-v c(s) g(s) Figure 5.4: Block diagram of feedback control These are considered in more detail in Section 5.3.2. Sensitivity, complementary sensitivity and loop transfer functions Consider a linear process model in terms of deviation variables as follows y = g(s)u + gd(s)d (5.5) where y denotes output variables, u the manipulated input and d a disturbance. g{s) and gd{s) are transfer function models which describe the effect on the output of the input and the disturbance. These are the basis of the controllability measures. Controller error is defined as e = r — y (5.6) where r(s) denotes the reference value (setpoint) for the output. Typically, some form of feedback control will be used to control the process, defined as: u = c(«)(r - y) (5.7) c(s) is the transfer function of the controller. If these equations are combined together and eliminate u the following closed loop transfer function is formed: y = Tr + Sgdd (5.8) CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 100 e = - S r + Sgdd (5.9) Here the sensitivity function 5 = (1 + gc)~^ and the complementary sensitivity function T = gc(l + gc)~^ = I — S. The transfer function around the open loop is L = gc. Scaling So that the controllability measures can be interpreted and meaningful comparisons made, it is necessary to express process models in terms of scaled, or normalised, variables. Thus all the input, disturbance and output variables should have a magnitude less than 1 [i.e. within the interval between -1 and 1). Skogestad (1996) indicated the following scahng m ethod: u = I'^'max where u' denotes the unsealed and u the scaled variable, and the largest allowable input change. For disturbances d = d'/d'^^^, for error, e = e'/e'max^ for output y = 2/V^maa; for reference, r = r'/ where d'^^^ is the largest expected disturbance and is the largest allowed control error. In most cases, it can be assumed that the maximum values {u'^ax->^'max^ &re independent of frequency. Controllability measures The following controllability rules are taken from Skogestad and Postlethwaite (1996) where they identified that it would be useful to quantify these reasonable heuristics for control system design. Let u>B denotes the bandwidth of the system, defined as the highest frequency where \L{j‘^B)\ = 1 and ojd denotes the frequency at which \gd{j^d)\ first crosses 1 from above. R u le 1 Ensure that speed of response is fast enough to reject disturbances. It is necessary th a t ÜJB > R u le 2 Ensure that speed of response is fast enough to follow setpoint changes. It is necessary that ujb > where tOr is the frequency up to which tracking is required. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 101 |L | Margin to stay within constraints (|u| Margin for performance (|y| 1 Control needed to reject disturbances z/2 Margins for stability and performance Figure 5.5: Controllability requirements R u le 3 Ensure that input constraints are not violated when perfectly rejecting distur bances. Must require that \g(juj)\ > at frequencies where \gd{jco)\ > 1. R u le 4 Ensure that input constraints are not violated for perfect setpoint tracking. Must require that \g{jco)\ > Rmax up to frequency where tracking is required. R u le 5 Ensure that speed of response is slow enough to allow for time delays in process and measurement. So with time delay 6 it is required that wg <1/0. R u le 6 Ensure that speed of response is sufficiently slow to allow acceptable control performance at low frequencies where a real RHP-zero, z, exists in the process. It is necessary th at wg < z/2. R u le 7 Phase lag constraint. Must require that wg < ujy,. Here the “ultimate” frequency, uju is where the phase oi g[juj) is —180°. NB. Similar to Bode stability criterion. R u le 8 Need sufficiently high feedback gain to stabilise the system where a real open-loop unstable pole in g{s) exists at s = p. The condition wg > 2p needs to be satisfied for stabilisation. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 102 5.3 Methods for controllability analysis In this section, methods are described for the performance of both the simulation and fre quency response controllabiUty analysis described earlier. A robust method for generating linear process information for numerically unstable models is also described. 5.3.1 Method for simulation controllability analysis In order to perform the simulation approach a process model, typically non-hnear is re quired. A candidate control system needs to be implemented and the controllers tuned by some method, e.g. Ziegler-Nichols. The following procedure would then be performed: 1. Simulate the process undisturbed to point of interest. If the process is batch then this would be one of a series of times throughout the batch. If the process is continuous it would typically be to the operating steady state. 2. Apply set-point change or process disturbance 3. Monitor the tracking error in the controllers until process restabilises 4. Calculate controllability measure such as an integrated error, described earher in Section 5.2.1 The procedure would be repeated for further times for batch processes. For both batch and continuous processes the procedure would be repeated for a range of disturbances and set-point changes. 5.3.2 M ethod for frequency response controllability analysis In order to use the frequency response techniques and other linear control tools it is necessary to generate a linear approximation to the uncontrolled process at the desired CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 103 point of linearisation. The linear process model has the following form: X — A x + 3 u y = Cx + T)u (5.10) The mathematics of how the state-space model is generated from the set of non-hnear differential and algebraic equations (DAEs) that form the uncontrolled process model (f{x,x,y,u) = 0) is set out in Appendix B.l. The transfer function representation of the linear model, used in Section 5.2.2, is generated by taking the Laplace transform of the state-space model. The inputs (u) and outputs (y) expressed in the linear state-space model are subsets of those in the non-hnear process model. The inputs (u) are selected as the manipulated variables and process disturbances. In this work, these would include such quantities such as flowrates (distillate and reflux), condenser duty and reboiler duty. The outputs (y) are selected as the controlled and measured process variables, such as composition, hquid level, and pressure. The foUowing procedure would be followed to generate the hnear state-space models of the process: 1. Simulate process undisturbed to predetermined hnearisation point with controhers if necessary 2. Remove control if present. Then extract hnear process information 3. Scale the hnear process model (Appendix B.2) 4. Apply controUabihty tools to hnear models. 5. Repeat for further hnearisation points if required 5.3.3 Robust method for linearisation The aforementioned method has proved to be adequate for simple equihbrium stage models and the packed column model. However, numerical difficulties have been encountered with CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 104 0.01 nonlinear - rigorous 0.009 linear - rigorous — linear - simple 17 0.008 0.007 % 0.006 Q 0.005 0.004 < 0.003 ÜJ 0.002 0.001 0 20 40 60 80 100 Time [si Figure 5.6: Comparison of robust method(Simple) to standard method(Rigorous) the rigorous equilibrium stage model. This section outlines those problems and proposes a method for dealing with these problems. The standard method outlined in the previous section was applied to the ethyl acetate case study presented in Chapter 3 modelled using the rigorous equilibrium tray model. The lin earised approximation to this model is compared to the original (non-linear, uncontrolled) model. As already discussed, it is expected that following a step change disturbance, these model responses should be approximately the same, for a reasonable period, which should be at least long enough for control to be effective. In Figure 5.6 the distillate composi tion response following a unit step increase in the reflux flow is shown. As can be seen from the flgure the linearised model response is initially in the correct direction but after 20 seconds, the response inverts and travels in the opposite direction. Clearly numerical problems in the linearised model result in this response. In order to resolve these issues, a modiflcation to the procedure is proposed. That is to CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 105 use the rigorous model to generate the state information and then to introduce this into the uncontrolled simphhed model, presented in Chapter 3 for the purpose of linearisation. The simphhed model has two important simphhcations compared to the rigorous model. 1. The pressure is assumed to be constant. 2. The dynamic energy balances are converted to algebraic ones where the hquid en thalpy is assumed locally constant. These assumptions are locally correct for short periods and are numerically stable. The results of applying the method are also shown in Figure 5.6. As can be seen a much closer agreement is obtained over the time period shown. In summary, this modihed hnearisation method can now be described as: 1. Using the rigorous model, simulate until the desired point of hnearisation 2. Initiate the simphhed model with the process conditions of the rigorous model at the point of hnearisation 3. Linearise the simphhed model at this point and then scale hnear model 4. Investigate the controhability of the process based on the hnearised version of the simphhed model 5. Repeat steps 1-4 for further hnearisation points within the batch It may be necessary to include controhers in the rigorous model to reach the desired process conditions prior to hnearisation. As long as these controhers are not included in the simphhed model, their effect wiU not inhuence the controUabihty. (The conditions at the point of hnearisation are, nevertheless, a result of the controhers.) The justihcation for using the rigorous model at ah is that there cannot be any conhdence that the simphhed model would be able to achieve the same process conditions as that of the rigorous model which was demonstrated in Chapter 3. Consequently, the hnearised models generated could be quite different. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 106 g 0.95 ■ 6 TRAYS 0.9 8 TRAYS ' 10TRAYS 0.8 0.75 Tim e [hr] Figure 5.7: Internal reflux ratio profiles 5.4 Controllability of batch distillation columns In this section, the controllability of both tray and packed batch columns is considered. Both the frequency response methods and simulation methods are used to investigate the effect of column type, column height and batch time have on the controllability of the column. The direct effect of the reaction in reactive batch distillation is also considered. 5.4.1 Case studies For the controllability analysis, six different column configurations are considered: the tray column, with 6, 8 and 10 trays and the packed column, with 4, 6 and 8 metres of packing. As indicated earher, due to the non-flnearities of the model and the changing process conditions, it is necessary to generate linear models at a number of time steps within the batch. Four time steps were chosen: 3 hours, 4.5 hours, 6 hours and 7.5 hours. All of these time steps are within the production phase when distillate is being withdrawn from the column. The ethyl acetate case study as presented in Chapters 3 and 4 is considered. The columns are aU operated under controlled composition, to 0.6 mole fraction of ethyl acetate. The reflux profiles for the three tray columns is shown in Figure 5.7 For the tray column, the linear models are generated using the Robust linearisation proce dure, described earlier. The packed column is linearised directly from an uncontrolled col umn. The linearisation was performed using the LINEARISEtd^sk within ^rPROMS which CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 107 Input Nominal U'^ m ax Ui Lq 2.15 mol/s ±2.15 mol/s 1/2 ^ 0.383 molf s ±0.383 mol/s U3 Qreb 8.85 X 10"^ J/s ±8.85 X 10^ J/s Output Nominal y max yi M t 100 mol ±10 mol 0.6 ± 0.02 2/3 Treb 346 K ± 5 K Table 5.2: Inputs and outputs for linear model (6 Tray column at 3 hrs lin. point) generates the state-space matrices (A,B,C and D). The state-space information was im ported into Matlab (The Mathworks Inc., 1996) for analysis. 5.4.2 Scaling The linear models are scaled. When scahng the Hnearised models, the reflux flow, To, the distillate flow, D, and the reboiler heat supply Qj-eb are considered as the inputs/disturbances, u for the linear model. The outputs, y, considered are the condenser holdup, M j, the dis tillate composition, X£>, together with the reboiler temperature, Treb- The distillate and reflux flowrates which are to be used as manipulated variables are allowed to vary by 100%, actual values depend on the column type, size and batch time. The reboiler, normally the principal source of disturbance, is expected to not vary by more than 10%. For the out puts: the condenser holdup, M j, is allowed to vary by 10 %, ±10 mol. A small variance in the distillate composition is allowed ±0.02. The reboiler temperature is permitted to vary by ±5 K. The results are summarised in Table 5.2 for the 6 Tray column at the first Hnearisation point, 3 hours. 5.4.3 Linear models The Hnear models developed are first used to understand the effect of step changes in the inputs to gain some general insights into control. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 108 In Figure 5.8 is displayed the distillate composition response to 1 mol/s step increase in reflux flow, distillate flow and to a 1 X 10^ J/s, step increase in reboiler heat duty at 3 hrs for the 6 tray column. The gain is positive for the reflux flow as more liquid returns to the column, improving separation. The gain is negative for the reboiler as vapour flow through the column increases, while the hquid reflux remains constant, hence a decrease in the internal reflux ratio. Both of these inputs have a strong effect on the distillate composition. The step increase in distillate flow has no effect on the composition, this is due to the fact that this causes no change to the internal flows within the column, it only increases the rate at which the reflux drum drains. Therefore, the distillate flow can only affect the composition by interaction with the reflux flow being used to control the level in the reflux drum. In Figure 5.9 is shown the effects of a step change in the reflux flow, distillate flow and reboiler heat duty on the temperature of the reboiler at 3 hrs for the 6 tray column. In creasing the reflux flow causes the temperature to decrease due to the increase of high volatility components to the column. The step increase in reboiler heat supply causes an increase in temperature, due to the greater rate of evaporation of high volatility compo nents. Again, as with the composition responses, a step increase in distillate has no effect on the reboiler temperature as no internal changes occur. It is noted that all three inputs have minimal effect on the reboiler temperature. In Figure 5.10 is displayed the responses of aU three of the tray column linear models (6, 8 and 10 trays) to 1 mol/s step increase in reflux flow. The flgure shows the response of the 6, 8 and 10 tray distillation columns on the top, middle and bottom plots respectively. On each plot are the responses at each of the four hnearisation points. It is evident that for each of the columns, the responses become progressively larger with time. The final step response, 7.5 hours, is significantly larger than those at the previous time steps and this difference becomes larger with column height. It is thought that this is due, in part, to the lower holdup of material in the column towards the end of the batch. This explains why the larger columns have a larger response at the final linearisation point since they have been operating at lower reflux and hence at 7.5 hours, holdup in the column will be CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 109 lower. For the two candidate control systems DV and LV discussed in Section 5.1, using distillate to control the composition cannot be achieved without interaction with the level control loop where reflux is manipulated. Therefore the performance of the DV scheme depends heavily on the performance of the level controller. The DV configuration does have practical advantages over the LV configuration, partic ularly during start-up. While the column composition is increasing during total reflux, no distillate is withdrawn so the distillate composition controller would saturate to zero. Level in the reflux drum is controlled by manipulating the reflux flow. However, this pe riod is uncontrollable using the LV configuration, where reflux is used to control distillate composition and the distillate to control the level in the condenser. During startup, the reflux valve would be fully open but the distillate valve would be fully shut to maintain the level in the reflux drum. Consequently level control cannot be achieved and the reflux drum empties. 0.25 0.2 - ^ 0...... Heat Duty (+1x10* J/s) Distillate (+ 1 moi/s) Reflux (+1 mol/s) IŒ 0.1 h 0§ s rt 0.05 1 I 0 -0.05 - 0.1 100 300 Time [sec] Figure 5.8: Distillate composition response to step change in inputs 5. CONTEOI OF FFACT/VF BATCH COFFAfNS 110 0.1 -02 - 0 3 -0 4 -0 5 Heat Duty (+^x 10 J/s) Distillate (+ 1 mol/s) Rellux (+ 1 mol/s) 100 200 300 400 500 600 Time [sec] Figure 5.9: Reboiler tein[)eratiire response to step change in inputs 5.4.4 Fi'equency response based controllability In Figure 5.11 is displayed the frecpiency response of the three tray columns at the final linearisation point (7.5 hours). The critical frequency, where phase is —180°, is reached at 3.77 rad/s for all columns. There is a difference in phase lag profile between 0.1 and 1 rad/s between columns which may have an important effect on controllability if there is extra phase lag added to the system by dead time or from a PI controller. In Figure 5.12 is displayed the frequency response of the three packed column heights at the final linearisation point (7.5 hours). The phase lag is smaller in the low frequency region, 10“ ^ to 10“ ^ rad/s, than for the tray column. However, the phase lag increases more rapidly than for the tray column with the critical frequency, Wg, being reached at the lower frequency of 0.0545 rad/s for the 8m column, 0.0698 rad/s for the 6rn column and 0.0711 for the 4m column. This suggests that controllability gets poorer with increasing column height as the critical frequency gets lower. When considering the change in frequency response with time. Figure 5.13 shows the 10 Tray column and Figure 5.14 shows the 8m packed column. The responses are generally similar for each time step, however, there is a slightly larger phase lag and slightly larger CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 111 gain for the final time step which suggests that both columns become harder to control at the end of the batch. 5.4.5 Controller tuning In this section, the tuning of composition controllers for both packed and tray columns is considered using the linear models for the column with 10 trays and the 8 m packed column. Proportional controllers with integral action (PI) are used and tuned using the Ziegler- Nichols method. The scaled models are closed using proportional only (P) controllers and subjected to a unit (0.02 molfrac) decrease in composition setpoint. The gain is increased until oscillation occurs. The ultimate gain, Ky,, and ultimate period, Py, for the oscillations, together with the corresponding controller parameters, (K and r/) are shown in Table 5.3. As expected from the frequency response analysis, the tray column can take a much higher gain before destabihsing compared to the packed column. Column Kn Pu K TI Tray -9780 71 s -4401 59 Packed - 3 110 5 -1 .3 5 92 Table 5.3: Controller tuning parameters The step response of the controlled hnear models are shown in Figure 5.15 for the tray column and in Figure 5.16 for the packed column. The consequence of the packed column being harder to control is shown by the the longer setthng time, approximately 60 times longer than the tray column. Having developed the controllers for both the tray and packed columns, it is possible to investigate the open-loop controUabihty of these processes. In Figure 5.17 is shown the frequency response of the tray column. The composition control by reflux process is shown as |G|, the effect of the reboiler disturbance \Gd\ a-nd the effect of the open-loop is shown as \L\. Here the bandwidth of the disturbance, Wj, is 0.0038 rad/s and the controUer bandwidth, 4.53 rad/s. The disturbance bandwidth is the frequency up to which control is required to reject disturbances. This is very low compared to the bandwidth of CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 112 the controller, and therefore the controlled process can be expected to reject disturbances effectively. However, the gain and hence bandwidth, may need to be reduced in a real system to account for unmodelled dead time. The requirement is that the bandwidth is smaller than 1/6 where 6 is the dead time. In this case, the controller would be destabilised by more than 0.2 seconds dead time. In a real system, the bandwidth could be reduced to allow for dead time of up to 262 seconds and still reject disturbances. It should be noted that, due to the disparity between non-linear and linear systems, confidence in the low frequency region is not high. Therefore, in practice, a real system may not be capable of rejecting expected disturbances with this dead time. In Figure 5.18 is shown the frequency response of the packed column. Here, the bandwidth of the controller loop, ui is 0.034 rad/s. Control to reject disturbances is required up to 0.0046 rad/s so therefore the packed column controller will be able to reject expected disturbances. The composition controller for the packed column wiU be capable of dealing with a dead time of up to 29 seconds and by reducing the bandwidth it is possible to allow for a dead time of up to 216 seconds and still reject disturbances. 5.4.6 Non-linear model simulations Having investigated the composition controllability using the frequency response technique and developed controller settings for the columns, the controllers are tested by simulation of the non-linear, rigorous models for tray and packed columns. The simulation approach to controllability is also applied. Two aspects of controllability are investigated. Firstly, how controllability is affected during the course of the batch and secondly how the choice of the number of trays or how the choice of packing over trays affects the controllability. In order to make these comparisons, four column configurations are employed: 6 , 8 and 10 trays and 8m packing. Each column is simulated, undisturbed to each of four points in time: 3 hours, 4.5 hours, 6 hours and 7.5 hours. Then a 10% step increase in the reboiler heat supply, the maximum expected, is applied. The responses of the composition con trollers was monitored until the composition settled. The Integrated Absolute Error (lAE) and the Integrated Time Absolute Error (ITAE) (Shinskey, 1994) are used to investigate CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 113 the controllability. These values for the four column configurations are shown in Table 5.4, the final value for the packed column is unavailable as it failed to settle. As is reflected in the frequency response study, control is poor at the end of the batch for aU case studies and the packed column has poorer performance than the tray column. However, earlier in the batch the column appears to get harder to control during the high reflux period (Figure 5.7). This suggests that the flow of reflux has some effect on controllability. 6 Trays 8 Trays 10 Trays 8m Packing Time [hrs] lAE ITAE lAE ITAE lAE ITAE lA E ITAE 3 4.67 1600 6.81 2890 6.81 2890 17.3 14600 4.5 6.17 1840 7.79 2860 8.49 4360 18.53 14100 6 6.36 1870 8T4 3940 8.81 3700 18.96 15625 7.5 5.87 3610 12.1 17000 3120 8 x 106 - - Table 5.4: Integrated controller errors The controller error is shown in Figure 5.19 for the the 10 tray column and in Figure 5.20 for the 8 m packed column. These reflect a similar performance to the linear simulations, with the tray column settflng in 60 secs and the packed in approximately 1 hour. However, the frequency of the oscillations is much lower which may be due to mismatch between the linear and non-hnear models. Also shown in Figure 5.19 is an example of the poor performance at the end of the batch where at 7.5 hrs where the column does not settle within 60 seconds. 5.5 Effect of reaction In this section, the controUabihty of both reactive and non-reactive systems are investi gated for the tray column model. However, in order to compare the controUabihty of the two systems, their process conditions must be as similar as possible for a comparison to be informative. Hence the foUowing approach is taken: • The rigorous tray model is simulated with control and reaction to the 3.5 hour hnearisation point. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 114 • The Robust Method for Linearisation (Section 5.3.3) is used to generate the reactive linear model • From the same linearisation point as above, the reaction terms in the mass and energy balances are set to zero and the Robust Method for Linearisation is used to generate the non-reactive linear model For simulation controllability comparison, the rigorous tray model is simulated with control and reaction for 3.5 hours, this is the reactive non-linear model. The reaction terms are also set to zero in the non-linear model to create the non-reactive non-linear model. 5.5.1 Simulation controllability Figures 5.21 and 5.22 show the controlled responses of the column to a set point change of 0.01 in the setpoint of the composition controller for the reactive and non-reactive non linear models. Figure 5.21 shows the response of the controlled ethyl acetate distillate composition and Figure 5.22 shows the response of the uncontrolled reboiler temperature. The dashed One represents the response in the reactive system model and the solid line represents the response in the non-reactive system model. It can be seen that there is no significant difference between the reactive and non-reactive systems in terms of the distillate composition. There is a slight difference in the temperature response which reflects the fact that ethyl acetate is not being produced in the non-reactive system model and therefore the temperature rises faster as the ethyl acetate concentration falls. 5.5.2 Frequency response controllability The process gain as a function of frequency for the reactive and the non-reactive model systems are shown in Figure 5.23. There appears to be very little difference between the reacting and non-reacting systems. The two curves separate slightly at high frequencies but the process gain is very small and therefore inconsequential. The phase is slightly different at low frequencies but otherwise similar (not shown). As already mentioned. CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 115 the further one moves from the point of hnearisation, the less r eh able the hnear model becomes. Therefore no meaningful conclusions can be drawn from this area of the response. For this case study it is concluded that reaction has minimal direct effect on control. However, as indicated earher, reaction does change the reflux ratio profile and therefore may indirectly affect control. It is beheved that for other kinds of reactions, for example highly exothermic or those with tight temperature requirements, control wih be more profoundly affected by reaction. 5.6 Conclusions In this chapter, the controhability of reactive batch distiUation columns was investigated using, both frequency response techniques and simulation based techniques. The frequency response techniques require the generation of hnear models. It has been concluded that packed columns are harder to control than tray columns. Increasing the height of a packed column makes the column harder to control, but increasing the number of trays has a much less significant effect on controhabihty. Control becomes harder for both types of column towards the end of the batch as the holdup lowers and the reflux ratio increases. It has also been concluded that, for this specific case study, reaction does not have a significant effect on control. However, due the distinctive reflux ratio profile in a reactive column, as compared to a non-reactive column, the reaction does indeed make the column harder to control. CTQ 5 rv Cn Ethyl A cetate C om postion Ax Ethyl a ce ta te com position A x Ethyl A cetate C om position A x O O o o o g d 3 to > % H O O C e: CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 117 10' 6 Trays 8 Trays — 10 Trays 10 10" 10" -90 -136 -225 -270 Frequency [rad/s] Figure 5.11: Tray column frequency response of distillate composition to reflux flow at 7.5 hrs 4m Packing 6m Packing 8m Packing 0 -9 0 i -3 6 0 -4 5 0 -5 4 0 10' 10' 10 10 10' Frequency [rad/s] Figure 5.12: Packed column frequency response of distillate composition to reflux flow at 7.5 hrs CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 118 3 hrs 4.5 hrs — 6 hrs I. — 7.5 hrs -90 -135 S’ -180 £ -225 -270 L - 10-^10' 10 10® Frequency [ratfs] Figure 5.13: Frequency response of distillate composition to reflux flow for 10 tray column (0 10 0 -9 0 0) -2 7 0 qI -360, -4 5 0 10" 10' 10 10’ 10" Frequency [rad/s] Figure 5.14: Frequency response of distillate composition to reflux flow for 8m packed column .5. CONTZ^OI OF FFACT/VF BATCF COFFAfNS' 119 - 0.2 - 0 .4 - 0.6 I -0.1 Q. E < - 1.2 - 1 .4 - 1.6 -2 60 Tim e [sec] Figure 5.15: Tray column closed loop response to setpoint change (Linear model) - 0 .5 < - 1 .5 -2 - 2 .5 1000 2000 3 0 0 0 4 0 0 0 Time [sec] Figure 5.16: Packed column closed loop response to setpoint change (Linear model) CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 120 Frequency [rad/s] Figure 5.17: Magnitude composition response (10 tray column) Frequency [rad/s] Figure 5.18: Magnitude composition response (8m packed column) .5. CONTROI OF FFACTiVF BATCF COFFMNS 121 0 015 3 hrs 0.01 0 005 b LU 2 -0 005 5 -0 0 1 -0 015 - 0.02 - 0 025 20 30 40 Time |sec| Figure 5.19: Composition rontroller error in res])onse to 10% step increase in reboiler heat duty (Rigorous Tray Column) 001 0.005 -0.005 UJ 2 - 0.01 § ^ -0.015 -0 02 -0 025 -0.03 500 1000 1500 2000 2500 35003000 4000 Time [s] Figure 5.20: Composition controller error in response to 10% step increase in reboiler heat duty (8m Packed Column) at 3 hours CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 122 0.62 ^ 0.615 0.61 < 0.605 0.6 ^ 0.595 non-reactive reactive 0.59 0 5 10 15 20 Time [min] Figure 5.21: Distillate composition response of reactive and non-reactive tray columns 346 345.8 345.6 Ô 345.4 345.2 non-reactive reactive 345 0 5 10 15 20 Time [min] Figure 5.22: Reboiler temperature response of reactive and non-reactive tray columns CHAPTER 5. CONTROL OF REACTIVE BATCH COLUMNS 123 ,-2 - 4 0) .•t; O)c (0 - 8 -10 -2 Frequency [rad/s] Figure 5.23: Composition frequency response of reactive and non-reactive tray columns C hapter 6 Modelling and control of short-path columns This chapter considers the modelling of reactive distillation in short path evap orators. From the literature it has been identified that although this process is used industrially no work has been undertaken on modelling this process and it is therefore important to develop a model so that the process can be more fully understood. A dynamic model for reactive short path distillation is presented which may also be used for control studies. The model is demonstrated using a complex, industrially motivated example with thermally unstable reactants and products. The sensitivity of the process to changes in operating conditions and the control of the process is investigated. 6.1 Introduction Reactive distillation has been applied successfully in industry where large capital and en ergy savings have been made through the integration of reaction and distillation into one system. Reaction yields can be increased by the removal of volatile products from the reaction zone, pushing the equihbrium towards the products. However, if the materials 124 CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 125 involved are temperature sensitive, they will degrade when exposed to high temperatures for extended periods of time. Operating batch distillation under vacuum will decrease the distillation temperature but large residence times within the still may nevertheless result in thermal degradation. Furthermore, the degree of vacuum is limited by the pressure drop across the column. Short-path distillation addresses these issues. In these columns (Figure 6.1), a liquid feed is applied continuously or semi-continuously to the inside wall of a single, externally heated, tube. A rotating wiper spreads and moves the film perpen dicular to the flow, thereby avoiding hot spots in the liquid. A condenser in the centre of the evaporator ensures a short-distiUation path with minimal pressure drop, allowing a high degree of vacuum and large evaporation rates. Performing a chemical reaction within a short-path evaporator is an atypical use for the equipment. However, it is being considered by industry as an alternative to reactive batch distillation, such as described in Chapters 3 and 4, when dealing with reactions involving highly temperature sensitive materials. No work has been undertaken in the open literature addressing the modelling of reactive short-path distillation. Therefore it is important to develop a model so that the behaviour of these processes can be more fully understood. Secondly, it is likely, as with chemical reactors where highly exothermic reactions take place, that the process is difficult to control. Therefore it is important that the model developed is dynamic so that the controllability of the process can be investigated. In this chapter, a dynamic model for short path distillation is presented. The model is demonstrated using a complex, industrially motivated example with thermally unstable reactants and products. The sensitivity of the process to changes in operating conditions is investigated. 6.2 Modelling of short-path evaporators In this section, the modelling of short-path evaporators is investigated. As identified in the literature survey in Chapter 2, the following general modelling issues have been identified CVfAPTEff G. AfODEIIEVG AND CONTROI OF SDOFT-PATF COFDAFV^ 12G FEED CONDENSER HEATING JACKET WIPING SYSTEM VACUUM — COOLING RESIDUE Figure 6.1: Short-path evaporator for iioii-reactive short-path evaporators: 1. Modelling of heat, mass and momentum transfer within the evaporator and con denser films. 2. Modelling of the mass transfer from the evaporator film surface and through the vapour gap onto the condensing surface. 6.2.1 Modelling of film phenomena Liquid hydrodynamics The flow pattern on the evaporator surface is com])lex, resulting from a combination of the gravity assisted flow axially and that created by the action of the wiper blades. McKenna CJMPTE/f G. MODEIIJNG AND CONTAOI OF SDOFT-FATF C0FFMN5' 127 5 Blade Wall Bow Wave Direction of blade movement Figure (J.2: Top view of mixing bow wave (adapted from McKenna 1995^ (1995) indicated that a rolling bow wave forms in front of the blade, shown in Figure 6.2, providing the mixing action. Cvengros et al. (1995) modelled the film behaviour in a column with segmented wipers. The column was divided into a series of laminar sections joined by mixers of zero length to represent the column wipers. This compared well to experimental residence time dis tribution data. Generally, the flow is assumed to be either laminar, resulting in radial temperature and concentration gradients, or turbulent in which case there are, no radial variations. Micov et al. (1997) noted that the well mixed, turbulent, regime can be approached in a wiped film evaporator operating at high speed. However, at high evaporation rates, thermal and concentration gradients do exist and cannot be eliminated. It was demonstrated that during the turbulent regime, separation performance is better than in the laminar regime. In order to model the fluid dynamics of the film, the appropriate form of the Navier-Stokes cnpiation needs to be included in the model. This can be quite complex but when assuming a laminar film, the Niisselt form of the equation is sufficient, Micov et al. (1997). This is the approach adopted in this tliesis. It is assumed that the radial mixing does significantly affect the flow along the column in the axial dimension. CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 128 evaporator distillation condenser I W vapour 11 0 Figure 6.3: Temperature profile in a wiped film evaporator (Lutisan et al 2002) Temperature and concentration profiles Temperature profiles can be quite pronounced within the column with large effects on the performance of the column, necessitating an energy balance. Temperatures will tend to rise axially downwards as the higher volatility products are removed. Also, there are radial temperature differences, particularly when flow is laminar. Figure 6.3 shows a typical radial temperature profile Lutisan et al. (2002). In the laminar regime, the temperature profile in the film is commonly assumed to be linear radially, whereas in the turbulent regime it is assumed to be constant radially. As the industrial column under consideration is mechanically wiped, it is assumed that the mixture is well mixed in all dimensions other than axially. Therefore temperature is assumed to vary only axially and not radially. 6.2.2 Modelling of evaporation phenomena Kawala and Dakiniewicz (2002) indicated that there are three major regimes encountered in high vacuum distillation: Molecular Distillation, Intermediate Range and Equilibrium CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 129 distillation. Which regime prevails depends on the Knudsen number, K n = A/ho, which expresses the ratio of the mean free path of vapour molecules. A, to the distance between evaporator to condenser, Lq. As pressure decreases the mean free path increases and so does the Knudsen number, Kn. Molecular distillation (Kn > 10) Here the vapour molecules travel the distance between the evaporator and the condenser with practically no colhsions. Evaporation proceeds at the maximal rate given by the Langmuir-Knudsen equation: " ^2irM iRT W here ji is the rate of evaporation of component i {rnol/w?s) and P° is the saturation pressure of component i {Pa). This equation applies to both the evaporation rate from the evaporator surface and the re-evaporation from the condenser surface. Intermediate range (0.05 < Kn < 10) Here evaporation rate is reduced through vapour interaction, reducing the mean free path of the vapour molecules. The apparent rate of distillation in this range, Je , is described by the following: 3E - f y- ji (6.2 ) where / = [1 — (1 — F){1 — A; and n are experimental coefficients. F, is the surface ratio defined as: ^ = 6 ^ ( - ) de is the diameter of evaporation surface curvature. A number of authors,e.g. Lutisan and Cvengros (1995a)(1995b) and Batistella et al. (2000), have considered the treatment of the vapour phase in detail and have used Monte-Carlo CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 130 based simulations to determine the overall behaviour of the vapour phase during the intermediate range. Equilibrium distillation (Kn < 0.05) Here collisions are frequent and vapour molecules frequently return to the liquid. Thermo dynamic equilibrium exists at the liquid-vapour interface. The parameter / tends towards the value of the surface ratio, F, and the evaporation rate is given by the following equa tion: Je = F X ji (6.4) For apparatus with planar evaporation and condensation surfaces, the surface ratio, F, is 0.5 (Kawala and Dakiniewicz, 2002). Thus, as a result of intermolecular collisions, the probability of a gas molecule reaching the condenser is equal to the probability of its return to the evaporation surface. 6.3 Dynamic short-path distillation model Previous authors (Batistella and Maciel (1996), Nguyen and Le GofRc (1997) and Lutisan et al. (2002)) have considered steady-state models of non-reactive short-path distillation. In this work, a dynamic model is presented that also considers multiple reactions within the evaporator liquid film. It is assumed that due to the mechanical wiping perpendicular to the flow, variations in temperature and concentration due to reaction, evaporation and external heating occur only in the axial direction. Axial flow, driven by gravity, is assumed to be laminar and vacuum sufficiently high that molecular distillation occurs (Kn > 10). The model is implemented in the gPROMS process modelling system (Process Systems Enterprise Ltd., 1999) and is shown in detail in the following section. CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 131 EVAPORATOR CONDENSER I -FILM WALL^ I 1 Figure 6.4: Cross section of Evaporator 6.3.1 Modelling assumptions As a result of the literature review, in Chapter 2 the following assumptions have been made in the development of a short path distillation model: • Dynamic component molar balance • Dynamic energy balance • Laminar liquid hydrodynamics • Molecular diffusion Dynamic molar balance Component molar balances are used to describe the flow of material through the Uquid film on the evaporating surface of the column (shown in Figure 6.4 as given by Equation 6.5). The film is considered to be well mixed perpendicular to the flow of material downwards CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 132 therefore the material balance is performed over the cross-sectional film annulus. The first term on the right hand side of the equation represents the flow in and out of this annulus. The second term represents the flow of material out of the evaporator film towards the condenser surface, governed by the appropriate rate expression. The third term is the reaction term, modelled as homogeneous within the evaporator liquid film. The material balances are defined for each component and across the whole length of the column with the exception of the boundary at the top of the column. d M ^ d F ^ = i = l,..,Nc, VzG(0,Z] (6.5) i=i The condenser surface is modelled similarly except that the interfilm transport term Niz is positive and the reaction term is omitted. Dynamic energy balance The energy balance (Equation 6.6) considers the transport of energy through the film by material (first term on right hand side) and that removed by material evaporating from the film (second term). The energy balance also considers the heat produced/ absorbed by reaction (third term) and that supphed by the heating medium through the walls of the column (fourth term). fiuE — = -— -Y,Ni,,hl,-2wT,Sf'£TiAHf~Q Vze(0,Z] (6.6) i=l j=l Laminar liquid hydrodynamics Extremely complex fiow patterns exist within a wiped film evaporator, a combination of both gravity assisted axial fiow and radial and tangential flows due to wiping. However, for the purposes of calculating the film thickness, it is assumed that the flow patterns in both the evaporator and condenser films are laminar. Therefore the film thickness can be described by Nusselt’s equation. For the evaporator film this is; f#. = Z'y'gCff)' V. e0 [. z] (6.7 ) CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 133 Film thickness definition, ,E _ ^ T , z ^T,2 VzG[0,Z] (6.8) ^ ^pTTTe Molecular diffusion It is assumed that the pressure in the short path column is sufficiently low to permit the rate of diffusion to be characterised by the Langmuir-Knudsen equation. It is also assumed that there is no re-evaporation from the condenser. = 1.006 X 27rr^—= A â J = i = 1, N c Vz G [0, Z] (6.9) y2'KmiRT^ Ancillary equations In addition to the above equations the following constraints and definitions are included in the model: Mole fraction normalisation, Vz G (0, Z] Nc ^i,z — 1 (6.10) t=i Definitions, i = 1,.., Ac Vz G (0, z] (6.11) ( 6.12) CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 134 6.3.2 Boundary and initial conditions The conditions at the boundary at the top of the column, z = 0, for the evaporator surface are specified as being equal to the feed of the column. Here total molar howrate, composi tion and temperature are specified. For the condenser surface the same specifications are made, however the feed flowrate is specified as neghgible but non zero. The column is operated from steady state. Therefore the initial condition is that all time derivatives are zero, dX/dt = 0. 6.3.3 Numerical solution The model, a set of partial, differential and algebraic equations (PDAEs), is implemented within the gPROMS modelling language. Spatial variations are only considered in the axial direction. The method used for spatial discretisation is the Backward Finite Difference Method (BFDM). Orthogonal Collocation on Finite Elements Method (OCFEM), which is typically more computationally efficient for a given accuracy, proved to be numerically unstable. The effect of the number of intervals selected for both dynamic and steady-state simulations is discussed in Section 6.4.1. 6.4 Case study An industrially motivated complex reaction scheme is used to demonstrate the applicability of the model presented in the previous section. The scheme is, shown in Figure 6.5. Both the reactant. A, and the product, B, are thermally unstable, degrading to form products, D and E. The volatile by-product, V, wiU react further with the desired product B, producing C, thereby lowering yields of B. The forward reaction is strongly exothermic, causing the film temperature to rise down the length of the evaporator even in the absence of an external heat source. However, the volatile by-product, V, removes some of the heat produced, thereby hmiting the temperature rise. In this case study, no external heating CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 135 is required, i.e. Q = 0 in Equation 6.6. The column configuration and properties of the feed, pure A, is outlined in Table 6.1. Operating under these conditions, the composition profiles of components A and B and the temperature profile within the column are as shown in Figure 6.6. The composition of the desired components B rises steadily along the length of the column as the composition of reactant A decreases. Under these conditions, there is minimal production of the side product C as the volatile component, V, is evaporated from the film. The temperature of the film decreases slightly along the column, from an initial temperature of 353 K to 350 K as heat is being removed by the evaporating methanol. This reduction in temperature also ensures minimal thermal degradation of the reactant and main product. This base case will be used initially to investigate the degree of discretisation required for both steady-sate simulation and during dynamic changes as a result of disturbances. It is then used to explore how the way in which the column is operated, affects the yields of the desired product. Specifically, the effects of feed flowrate, feed temperature, heat of reaction and evaporation efficiency are considered. A B +V c N/ D E Figure 6.5: Reaction Scheme Column Length 0.4 m Evaporator Diameter 0.2 m Condenser Diameter 0.04 m Feed Flowrate 1 mol / hr Feed Temperature 353 K Feed Viscosity 0.1 Feed Density 1000 kg/m ^ Table 6.1; Short path column configuration CffAPTE7( 6. MODEIMNG AND CONTEOI OF 5VfOPT-PATE COEDAfNS' 136 — Component A • • " Component B £ 0.6 0.4 O 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Axial distance [m] 353 352 5 ^ 352 3 351 5 351 H 350 5 350 349 5 0 0 05 0.1 0 15 0 2 0 2 5 0.3 0.35 0.4 Axial distance [m] Figure 6.6: Base case profiles: compositions (top), temperature (bottom) 6.4.1 Effect of discretisation As indicated previously, for stability reasons, the method of discretisation employed in the conversion of the partial differential equations (PDEs) into differential equations is the backward finite difference method (BFDM). In this section, the degree of discretisation required for accurate representation is investigated. As the process is continuous, the performance under steady state is important. However, as the model is also required for control studies, it is important that the degree of discretisation is sufficient to represent the model during dynamic responses to disturbances. In Figure 6.7 (Top), the effect of a step increase of feed temperature of 5 K on the column is displayed. The resulting temperature profile at the outlet in shown for two levels of discretisation, 3 intervals and 20 intervals. It is clear that the two plots remain close throughout the transition from one steady state to another. The discrepancy between the two plots remains at less than 0.02%. Figure 6.7 (Bottom) shows the steady-state CHAPTER 6. MODELLING AND CONTROL OE SHORT-PATH COLUMNS 137 355 354 20 Elements 353 3 Elements 352 351 350 0 2 4 6 8 10 12 14 16 18 20 Time [tirs] 352 5 20 Elements -w - 10 Elements ? 352 -B - 5 Elements -A - 3 Elements 3 351 5 351 H 350.5 350 349,5 0 0 1 0.2 0.3 0.4 0 5 0 6 0 7 Length [m] Figure 6.7: Outlet Température profile resulting from disturbance (TOP) Steady-state temperature profile (Bottom) temperature profile across the column before the step increase of feed temperature. The temperature profile is displayed for 3, 5, 10 and 20 intervals. It is clear from the plot that the 3 and 5 interval discretisations do not give sufficient coverage of the interior of the column and consequently give different profiles. However, the 10 and 20 interval profiles are close with negligible difference and therefore it can be assumed that 10 intervals are sufficient to give good accuracy. This degree of discretisation will be used when assessing the sensitivity of the column to changes in operating conditions in the following. 6.4.2 Effect of feed Howrate The liquid film on the evaporator surface, as it is assumed to be well mixed perpendicular to the flow, shares many properties with a plug flow reactor, PER. However, there is an important difference between the two. The plug flow reactor has a fixed volume due to the CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 138 geometry of the reactor. The volume of a short-path evaporator film on the other hand depends on the depth of the film, 6, and the geometry of the evaporator. The film depth depends on the fiowrate of the liquid feed and its density and viscosity within the laminar regime. The residence time in a reactor is defined as r = F/jP, where V is the volume of the reactor and F is the volumetric flowrate of the feed. For the liquid film, the volume V is the product of the surface area of the evaporator cylinder, 27rr’eZ, and the depth of liquid, 6. Therefore the definition of space time becomes: 2'KTpZb T = (6.14) F where the film thickness, 6, is described by Bird et al. (1960) as: ' S/^F 6 = (6.15) pg27rr. Therefore for a PER, r oc F~^ but for the film reactor, r oc F~^. This is illustrated in Figure 6.8. The film reactor’s residence time decreases less rapidly with increasing fiowrate due to the increase in cross sectional area as the film gets thicker. 1600 PFR 1400 FILM 1200 800 600 400 200 1 1.5 2 2.5 3 3.5 4 Flowrate [mol/hr] Figure 6.8: Effect of feed flowrate on residence time The effect of the feed fiowrate is investigated for the case study for a range of fiowrates from 1 mol/hr to 4 mol/hr as shown in Figure 6.9. As expected, the yield decreases, from 79% at 1 mol/hr to 38% at 4 mol/hr, due to the fall in residence time. It should be noted that the decrease in yield would be expected to be larger in a PFR due to the more rapid drop in residence time. CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 139 80 70 >: 50 40 30 2.5 3 .5 Flow [mol/hr] Figure 6.9: Effect of feed flowrate on reactor yield 6.4.3 Effect of viscosity The reacting mixture in the evaporator is assumed to have a constant viscosity of 0.1 PaS. Although in reality, the mixture viscosity would be temperature dependent. It is, however, important to investigate how the yield is affected by viscosity. In Figure 6.10 is shown how the yield varies as viscosity varies between 0.0001 PaS and 10 PaS. At the lowest viscosity, the yield of B, is only 33%. At the highest viscosity, the yield approaches 100%. As viscosity increases, the film thickness increases (see Equation 6.15) and so does the residence time, shown in Figure 6.11. At very high viscosities, outside the range of the figure, the yield begins to fall as the thermal degradation reactions become significant. 100 9 0 80 70 I 6 0 50 40 30 10'^ 10'' Viscosity [PaS] Figure 6.10: Effect of film viscosity on reactor yield CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 140 800 0 6 0 0 0 400 0 2000 10'^ Viscosity [PaS] Figure 6.11: Effect of viscosity on residence time 6.4.4 Effect of feed temperature The effect of feed temperature on the yield of the product is investigated and shown in Figure 6.12. If the feed temperature is too low then there is insufficient time for the product to be formed due to the slower rate of reaction. As the temperature rises, the rate of reaction rises and therefore the yield of product is higher. However, if the feed temperature is too high, then thermal degradation of both the product and the reactant wiU occur, thus lowering the yield again. This is observed in Figure 6.12 as a maximum yield of 99.6% is found at a feed temperature of 368 K, however, as the temperature increases beyond this, the yield is reduced and at 383 K, has dropped to 94.8%. This is due to the thermal degradation of the reactant A to D, and of the desired product, B to E. 100 95 85 80 75 35 0 355 360 365 370 375 380 385 Feed Temp [K] Figure 6.12: Effect of feed temperature on reactor yield CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 141 6.4.5 Effect of heat of reaction The effect of uncertainty in the heat of reaction is found to be quite dramatic. For the base case, the heat of reaction is initially assumed to be identical to the heat of vaporisation of the volatile by-product. The heat removed by the evaporation of the volatile by-product balances the heat produced by the main reaction. A 1% increase in the heat of reaction results in a slight warming of the reaction mixture over the length of the evaporator and consequently the yield increases (from 79% to 87%). Lowering the heat of reaction by 1% causes a significant drop in temperature across the reactor resulting in a decrease in yield (from 79% to 70%). It is interesting to observe that the process is extremely sensitive to the heat of reaction and that for simulations, small uncertainties wiU have a large effect on the predicted yield of the reactor. 6.4.6 Effect of efficiency So far, it has been assumed that evaporation takes place without resistance in the vapour phase and without re-evaporation from the condenser. The impUcations of lower mass transfer rates are now considered. For non-reactive evaporation, the lowering of evapo ration efficiency, compared to molecular distiUation, significantly reduces the yield of the desired product (Batistella et ah, 2000). In the reactive case considered here, the effects are more subtle. The reduction in efficiency as a result of resistance in the vapour phase, for example at higher pressures, has a number of effects. Firstly, the equilibrium con centration of the volatile by-product, V rises in the Uquid establishing a new equiUbrium concentration. The higher volatile by-product concentration will increase the rate at which the desired product is reacted further in the side reaction to C. The degree to which this occurs depends largely on the rate constant of the side reaction as weU as on the efficiency of the evaporation. However, for this case study, these phenomena are significant for only very low efficiencies (< 1%) and therefore the column may be operated over a large range of efficiencies without significant reduction in yield. In Figure 6.13 is shown the effect when efficiency drops to zero and separation of the volatile CJfArTER G. MODEII/NG AND CONTJ^OI OF SZfORT-FATN COFFMNS 142 component, V, stops. Here the mole fraction of the desired product, B, is almost negligible, 0.0068, whereas the mole fraction of the by-product, C has risen to 0.69 at the outlet. The reason is that the desired product is consumed by reaction with the volatile component, V, in the film by forming by-product, C. This last example provides an important motivation for operating this reaction in an evaporator as it would be impossible to produce the desired product, B, without separating off component V. V ■ - • Component A fo.8- — Component B Component C l o . e - I 0.4 - E O 0.2 - 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Axial distance [m] Figure 6.13: Evaporator composition profile with no separation of V 6.5 Control of short-path distillation In this section, the control of short-path distillation is considered. The case study (Sec tion 6.4) established that the composition of the desired product, B, is particularly sen sitive to feed flowrate, heat of reaction and to the temperature of the feed. In order to control the outlet concentration of the evaporator, the feed flowrate is the only variable that can be manipulated. The other two variables: heat of reaction and feed temperature are considered as disturbances to the system. In this case study, where no external heat ing is applied, the heat supply is discounted as a source of disturbance or as a potential manipulated variable. The frequency response approach, outlined in Chapter 5 is used to investigate the controllability of the process. CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 143 6.5.1 Linearisation As the column operates continuously, the operating base case steady-state shall be used as the linearisation point. Figure 6.14 (TOP) shows the dynamic response of the non linear process model to step changes in the input variables. The step changes are a 0.5% increase in the heat of reaction, a 10% increase in the feed flowrate of A and a, -\-l K change in the feed temperature. New steady-states are achieved, about 12 minutes after the step changes are apphed. As reflected in the case study, the increases in the heat of reaction and the feed temperature both cause an increase in the composition of the desired product, B. The increase in feed flowrate, as expected, causes a decrease in the product composition as residence time falls. In Figure 6.14 (BOTTOM) is shown the same step changes for the linear version of the model. It is clear that the linear model provides a good approximation to the non-linear process model and it can be used with confidence in the control study. However, it should be noted that the linear model will not account for the decrease in composition of the desired product B, encountered for large feed temperatures when thermal decomposition becomes significant. 6.5.2 Frequency analysis The linear model is scaled with the output composition allowed to vary by, ±0.01 mole fraction. The manipulated variable, feed flowrate of A, is allowed to vary by, ±10%. The maximum expected disturbances are ±0.5% in the heat of reaction and ±1 K for the temperature of the feed. The scaled model is used for the frequency analysis. In Figure 6.15 is shown the frequency response of the composition of product, B, to the feed flowrate. It is clear that there is a very large phase lag in the process due to the high order of the process. This will have a destabilising effect and limit the gain of any implemented controller. CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 144 6.5.3 Controlled response For the control, the feed flowrate is manipulated to maintain the column composition of the desired product, B. Feed temperature and heat of reaction are considered as distur bances. A proportional integral controller is implemented and tuned using the approach of Ziegler and Nichols (1942). The controller gain, K c is 0.25 and the integral time, tj is 12.5 min. The frequency response of the gain of the open loop (process and controller) is shown in Figure 6.16. The response of the feed temperature disturbance is also shown on the figure. The difficulty in controlling this process is indicated by the fact that the distur bance bandwidth, Wj at 0.54 rad/min is above that of the controller bandwidth, at 0.05 rad/min. Therefore this controller is not effective at meeting the requirements of rejecting disturbances while maintaining the product within the required limits during its response. The linear models are simulated with control for a set-point change in Figure 6.17 and a feed temperature disturbance in Figure 6.18. The response to a disturbance in heat of re action is approximately the same as that for the temperature disturbance. The controller response to both the set-point change and the disturbance case is very slow, taking over 100 minutes to settle. The conclusion from Figure 6.16, that the controller is ineffective against disturbances, is shown by the fact that the output concentration deviation be comes 3.3 times the permitted maximum during the response. Some form of feedforward control may be necessary to meet the product tolerances specified. It can be concluded that the control of these processes is difficult due to the large process time lag along the length of the column and the sensitivity of the system to small dis turbances in the heat of reaction and column feed temperature. The column will become uncontrollable at high temperatures with the feedback control scheme described when the composition of desired product, B, decreases due to thermal degredation (Figure 6.12). CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 145 The action of the controller would be to decrease flowrate, which would increase the de gree of thermal degredation. Therefore some form of temperature control, in addition to composition control may be necessary. 6.6 Conclusion The objective of this chapter was to develop a dynamic model that describes the behaviour of reactive distillation in a short-path distillation column and use that model to investigate the control of this process. The model was demonstrated using an industrially motivated reaction. The reaction contained a thermally unstable reactant and desired product, making it suitable for treatment in a short-path distillation column under vacuum where both temperatures and contact times are low. The effect of feed flowrate, feed temperature, heat of reaction and evaporation efficiency were considered. The impact of these variables on product yield is significant except for evaporation efficiency, where a significant effect is only encountered for very low efficien cies. Thermal degradation effects were minimised by operating at low temperature and low residence times, and yields enhanced by the removal of volatile by-products, which successfully demonstrated the suitability of short-path distillation for combined reaction and separation. However, the control of this process is difficult. The sensitivity of the pro cess to operational changes, especially in feed temperature and heat of reaction, and the long time lag in the column contribute to this difficulty. The composition control of this process is made more difficult by the reactions as the trend of increasing desired product composition with temperature reverses at high temperature due to thermal degradation. Therefore good composition and temperature control of this process is critical. CHAPrER 6. MODELLING AND CONTROL OE SHORT-PATH COLUMNS 146 0.05 0,04 0,03 0,02 0,01 Q. A,IN - 0,01 - 0,02 -0,03 10 T im e [m in] step Rœponse 0,05 0,04 0,03 0,02 g 0,01 A,IN - 0,01 - 0,02 -0,03 0 2 4 6 8 10 12 T im e [min] Figure 6.14: Product, B, coinposition step responses. Non-linear model (TOP), linear model (BOTTOM ) CVfAf TEA 6. AfODEIITNG AND CONTAOI OF ^EOAT-TATN COIDAfNS 147 10' 10' 10 ' 720 ID -720 - -1440 -2160 10 Frequency [rad/'min| Figure 6.15: Short-path coluinii fretpiency response of product B composition to feed fiow -2 -3 10 Frequency [rad/min] Figure 6.16: Short-path column freriuency response of controller loop and temperature disturbance CHAPTER 6. MODELLING AND CONTROL OF SHORT-PATH COLUMNS 148 0.8 0.6 ■S .2 Q. E < 0.2 - 0.2 80 90 100 Time [min] Figure 6.17: Column composition response to a set-point change (controlled scaled linear model) 3 .5 2 .5 I Q. I 0 .5 30 100 Tim e [min] Figure 6.18: Column composition response to a feed temperature disturbance (controlled scaled linear model) C hapter 7 Conclusions and directions for future work This chapter summarises the main findings of this thesis regarding the mod elling and control of reactive batch distillation in tray and packed columns and the modelling of reactive distillation in short-path evaporators. Some directions for future work are also given. 7.1 Conclusions From a survey of the literature (Chapter 2) it was established that the majority of the previous work on reactive batch distillation has focused on modelling with some work on optimisation of operating policies and some Umited work on control. Some authors have considered control, for example, Reuter et al. (1989), Wilson and Martinez (1997a), Wilson and Martinez (1997b) and Monroy-Lopereba and Alvarez-Ramirez (2000) but they have tended to use simple models. It is also noted that with, the exception of Sprensen and Skogestad (1994) and Sprensen et al. (1996), no work has been undertaken on the controllability of reactive batch distillation. Analysis of the controllability forms the basis for understanding the features that make control of these processes difRcult. Adequate 149 CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 150 control is essential for maintaining consistent operation. The flexibility which is offered by batch operations can be further enhanced by the knowledge of how to modify the process design, its operation or control structure to yield better controller performance. In the literature survey, it was also concluded that no work has been undertaken on the modelling of chemical reaction in short-path columns or on the control of these processes. The main objectives of this thesis were to study the controUablity of reactive distillation in tray and packed batch columns and in short-path distillation columns. In order to study control, it is essential to develop rigorous dynamic models that accurately capture the process behaviour. In Chapters 3 and 4, the modelling of tray and packed columns were considered. In Chapter 5, the control of these columns was investigated and in Chapter 6 the modelling and control of reactive short-path distillation columns were considered. 7.1.1 Modelling of reactive batch distillation in tray columns The objective of Chapter 3 was to present a rigorous dynamic model for the simulation of reactive batch distillation. A simplified, but more numerically robust, model was also presented for use in the control studies in Chapter 5. In order to justify the adoption of the rigorous model over the simplified model for simulations, the two modelling approaches were compared for the production of ethyl acetate. Two operating policies: constant reflux ratio and controlled distillate composition, were considered for the production of a 0.6 mole fraction mixture of ethyl acetate and simultated using both models. The controlled policy was also considered with reboiler disturbances. In comparing the two modelling approaches, it appears that, although the predicted batch holdups are reasonably close, less than 5% discrepancy between the methods, the dis crepancies in the predicted batch times are significant and they increase as the column is operated under disturbances. For the constant reflux ratio policy, the reflux flow re mains approximately constant throughout. For the controlled case study, the reflux flow is changed continuously. For the disturbance case study, the reboiler disturbances intro duce greater variances in the reflux flow. This large difference in the performance of the simplified model compared to the rigorous under control would justify the adoption of the CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 151 rigorous model for simulating reactive batch distillation. As reported by Sprensen and Skogestad (1994), reactive batch distillation cannot be effectively operated without some form of feedback control. The simulation times for the two modelling approaches were markedly different. The simplified model took just 36 s but the rigorous took 574 s to solve, approximately 16 times longer. The sixteen fold increase in computational time is particularly significant if the rigorous model were to be employed for optimisation. When using a feasible path approach for dynamic optimisation, a complete simulation is required to evaluate the ob jective function at each step in the optimisation method. However, Monroy-Lopereba and Alvarez-Ramirez (2000) indicated that the controlled composition study can be thought of as being equivalent to the optimal reflux ratio policy. It is important to note that the “optimal” reflux ratio profile for the rigorous model may be very different to that predicted for the simple model. 7.1.2 Modelling of reactive batch distillation in packed columns In Chapter 4, the production of ethyl acetate in a packed reactive batch distillation column was considered. Packed columns can either be modelled as an equivalent tray column by determining a suitable value for the HETF (Height Equivalent to a Theoretical Plate) or more physically realistically by a rate-based model. The objective of this chapter was to establish whether or not the less physically realistic tray column model would be suitable for modelling the packed column for reactive purposes. A packed column model which extended the work of Furlonge (2000) to include chemical reaction throughout the liquid phase, was presented. This model was used to simulate the behaviour of the constant reflux ratio and controlled composition case study presented in the previous chapter but now, in a packed batch column. A method was presented for establishing the HETP for a batch column and this was used on both operating policies: constant reflux ratio and controlled composition. It was concluded that the HETP is not constant, varying with both time and packing height as a result of varying composition CHAPTER?. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 152 and liquid flowrate. Consequently, the controlled composition policy was found to vary more than the constant reflux ratio pohcy due to greater changes in liquid flowrate. A comparison was made between the packed column model and the rigorous and simpflfied tray column models with 10 trays, which corresponded to the average HETP found for both the constant reflux ratio policy and the controlled composition policy. The packed column was also compared to rigorous tray columns with 14 and 6 trays which correspond to the maximum and minimum HETP values for the controlled composition policy. The rigorous column model with 10 trays agreed most closely with the packed column model while the simplified model and the larger and smaller rigorous column models had poorer agreement. This suggest that if the equilibrium approach is taken, then it is vital to use a correct HETP value and to model the column using the rigorous and not the simplified, tray column model. However, for the controlled case study with and without disturbances, the constant HETP assumption begins to break down due to the higher variance in HETP experienced within the packing due to the varying flowrates. The greater variance may also be due to the different dynamics within the columns leading to a slower behaviour of the packed column. The variation in HETP was found to be too great to justify the adoption of the equilibrium approach for modelling packed columns especially where liquid flowrates vary considerably, such as during control. Therefore, in order to study control and controllability of packed columns it is necessary to adopt a modelling approach that accounts for the physically different mechanisms of heat and mass transfer encountered in a packed column and the rate-based packed column model will therefore be adopted when investigating packed column control. 7.1.3 Control of reactive batch distillation In this chapter, the controllability of reactive batch distillation columns was investigated using, both frequency response techniques and simulation based techniques. The frequency response techniques require the generation of hnear models. It was concluded that packed column are harder to control than tray columns. Increasing the height of a packed column CHAPTER?. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 153 makes the column harder to control, but increasing the number of trays has a much less significant effect on controllability. Control becomes harder for both types of column towards the end of the batch as the holdup lowers and the reflux ratio increases. It has also been concluded that, for this specific case study, reaction does not have a significant effect on control. However, due the distinctive reflux ratio profile in a reactive column, as compared to a non-reactive column, the reaction does indeed make the column harder to control. 7.1.4 Modelling and control of reactive short-path evaporators The objective of this chapter was to develop a dynamic model that describes the behaviour of reactive distillation in a short-path distillation column and use that model to investigate the control of this process. The model was demonstrated using an industrially motivated reaction. The reaction contained a thermally unstable reactant and desired product, making it suitable for treatment in a short-path distillation column under vacuum where both temperatures and contact times are low. The effect of feed flowrate, feed temperature, heat of reaction and evaporation efficiency were considered. The impact of these variables on product yield is significant except for evaporation efficiency, where a significant effect is only encountered for very low efficien cies. Thermal degradation effects were minimised by operating at low temperature and low residence times, and yields enhanced by the removal of volatile by-products, which successfully demonstrated the suitabifity of short-path distillation for combined reaction and separation. However, the control of this process is difficult. The sensitivity of the pro cess to operational changes, especially in feed temperature and heat of reaction, and the long time lag in the column contribute to this difficulty. The composition control of this process is made more difficult by the reactions as the trend of increasing desired product composition with temperature reverses at high temperature due to thermal degradation. Therefore good composition and temperature control of this process is critical. CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 154 7.2 Directions for future work In this section, some of the limitations of this work are highhghted and recommendations for future work are outlined. 7.2.1 Model validation This thesis has employed detailed models to explore reactive distillation in three unit op erations. The more detailed models are expected to have closer agreement to experimental data than simpler models used previously in the hterature. In order to verify the accuracy of the tray, packed and short-path column models presented in this work, experimental results are required. A close match between experimental results and model predictions would further justify the use of such a high level of modelling detail despite the associated large computational cost. 7.2.2 Modelling detail Experimental work such as that described in Section 7.2.1 may reveal the need for even more accurate process models, reaction models and physical property models. Process models The rigour of the tray column model may be enhanced by taking the following affects into account: • Tray Efficiency - For non-reactive distillation it may be possible to introduce Mur- phree tray efficiencies, however, as was indicated by Ruiz et al. (1995), no methods have been reported for the effect of reaction on plate efficiencies. • Entrainment Effects • Downcomer dynamics CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 155 For the packed column, the following effects may be introduced: • Mass transfer in the liquid phase • Dispersion effects Reaction models In this thesis, the reactions have been assumed homogenous. To gain further insight into the affect of chemical reaction, rigorous kinetics should be used. In the case of het erogeneously catalysed reactions, a thorough understanding of the transport mechanisms involved, is essential to model the reaction properly. Physical properties In this thesis, the physical properties have been assumed to be ideal. Real systems are typically non-ideal. To properly characterise the system it may be necessary to use more rigorous physical properties. The presence of azeotropes affects the feasibility of certain separations and may have implications for control. For the short-path column, it is assumed that pure molecular distillation occurs. It is also assumed that no resistance occurs in the vapour phase. At anything other than high vacuum, some resistance wiU occur. Some authors have considered modelling the affect of vapour phase resistance using Monte-Carlo simulations (Lutisan and Cvengros (1995a)(1995b) and Batistella et al. (2000)) 7.2.3 Further control studies Effect of reaction It was concluded that the particular reaction used in this thesis, did not have a significant affect on controllability. It would be useful to explore this further by considering a number of different reactions to establish whether or not this is a general conclusion or specific to the reaction chosen. CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK 156 Systematic approach to model reduction In this thesis, an approach was presented for avoiding numerical problems while generat ing linear models. The procedure used a simplified model. The degree of simplification necessary to ensure a numerically stable linear model, was established by trial and error. In this particular case, the energy balance was assumed to be algebraic and the pressure constant. However, it would be useful to develop a systematic approach for this model re duction to ensure that the minimum degree of simplification is applied while guaranteeing numerical stabihty. Nomenclature tte effective interfacial area per unit volume rn?jv n ? A tray area rr? Ab cross-sectional area of packed bed m ? A h total area of holes on a tray w ? Dout distillate flowrate m o lls Fyj francis weir formula wall interaction parameter - g acceleration due to gravity m/ 5^ h specific molar enthalpy J jm o l hyjeir weir height m hliquid level of liquid on tray m H enthalpy flowrate W heat of reaction j J jm o l k' heat transfer coefficient W jK m ^ K°'^ overall mass transfer coefficient m js Ki liquid-vapour equilibrium constant for component i - L liquid flowrate m o ljs Mb holdup per length of packed bed m o ljm Mb,i holdup of component i per length of packed bed m o ljm Mi molar holdup of component i mol Mtot total molar holdup mol N{ rate of transfer of component j from vapour phase to liquid phase m o ljsm P pressure Fa 157 NOMENCLATURE 158 Q reb oiler heat duty J /s Qc condenser heat duty J /s Rout reflux flow m o l/s Rint internal reflux ratio - Te radius of condenser m Te radius of evaporator m f'j rate of reaction j mol / m ^s T temperature K r* temperature at interface K n normal boiling temperature K Utot internal energy J V vapour flowrate m o l/s ^tray volume available between trays w? ^vessel vessel volume m? y vapour mole fraction mol jm o l y* vapour mole fraction at the liquid/vapour interface mol / mol z packing height m Z total length of packed bed m Greek Letters a pressure drop vapour flowrate parameter - 6 film thickness m € packing void fraction - l'ij stoichiometry of component i in reaction j - P average molar density m o l/m ^ P average mass density kg jm ^ fugacity coefficient NOMENCLATURE 159 Superscripts and Subscripts C condenser E evaporator NO number of components NR number of reactions L liquid phase V vapour phase in inlet stream out outlet stream Bibliography S. 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PROCESS MODELS 167 • Phase equilibrium • Perfect mixing • Immediate heat input • Negligible holdup in the condenser Molar balance on component, i: — Lin^i,in "P Vinyi,in Lout^i ^outVi "P ^ ^ ^ ] ^ij^j ^ — 1; Nq(^A.1^ Liquid and vapour contributions to component holdup: M ; = + M ^ o tV i i = 1, -/Va (A.2) Energy balance: rITJ f \ NR —= ^Li„hi in ^ fn ++ ^Vir,hJ„ in ^ Y n ~ - ~ Voutf^^ + ( - (A .3 ) i= i Vapour and liquid contributions to internal energy: Utot = M tth ^ + MYoth^ - P V tray (A.4) Total volume constraint: Vtroy (A.5) Equilibrium relationship: y I — R{X{ z = 1, N Q (A. 6) APPENDIX A. PROCESS MODELS 168 Normalisation equations: NC NC Y^Xi = J2yi = '^ (A.7) î=l i=\ Liquid Hydrodynamics: hliquid = (A.8) = 1.776488L»./ (A.9) Vapour Inflow Characterisation: AP - Pin - P - p ghiiquid (A .10) V^n = sqrt I j pYnAh (A .ll) \P in ^ / APPENDIX A. PROCESS MODELS 169 A.2 Rate-based model of packing section This is the rate-base model for the packing section of the column. This model extends the work of Furlonge (2000) to include reaction in the hquid phase. Reaction terms are added to the liquid phase component mole balance and energy balance. L(z=Z) m m z = Z Packed Column Section z = 0 V(z=0) m m Figure A.2: Packing section Assumptions: • Film theory for heat and mass transfer • Negligible hquid phase mass transfer resistance • Neghgible axial and radial dispersion in both hquid and vapour phases • Adiabatic operation • Liquid phase chemical reaction APPENDIX A. PROCESS MODELS 170 Boundary Conditions Liquid Inlet: Lin = L{Z) (A.12) Xi,in = ^i{Z) * = 1> ", (A.13) = H^{Z) (A.14) Vapour Inlet: Vin = y(0) (A.15) ÿi.TO = 2/i(0) 1 = 1 ,.., A c (A .16) V in h l = A'^(O) (A.17) Liquid phase Vz G [0, Z) Molar balance on component i: ^ = ^ + ^•' + (^) g i = 1,.., Ac (A.18) Energy balance: ^ - + k '^ a M { T * - T ^) + E ^ ^ ( - A i 7 f )r,- (A. 19) i = l \ P / 7 = 1 Mo/e fraction normalisation: Nc Y ^X i^l (A.20) i = l APPENDIX A. PROCESS MODELS 171 Vapour phase Vz G (0, Z] Molar balance on component i: ^ = i = (A.21) Energy balance: f)TjV r)TT^ = - — - - T") -'£N ihY(P,T’) (A.22) i=l Mole fraction normalisation: Nç = 1 (A.23) i=i Liquid/ vapour interface Vz G [0, Z] Mass transfer rate of component i IZj ~ ~ ~ (a.24) p = a^A}) j^rj^y {Vi ~ V i) * — I 5 ••? ~ 1 Energy balance: Nç Nç k '^ a ,A i( T ' - T^) + ^ A./if(P,T^) = k'''a,At{T'' - T ) NihJ{P,T^) (A.25) i=l Î=1 Phase equilibrium: 4{P,T\x)xi = Mo/e fraction normalisation: Nc Eÿ.* = l (A.27) i=l APPENDIX A. PROCESS MODELS 172 Auxiliary equations Vz G [0, Z] Definition of enthalpy flowrate: = L h ^ (A.28) = VhX (A.29) Definition of component holdup: = M^Xi i = 1 , Ac (A.30) Pi i = 1, Nc (A.31) Internal energy: u t = Mth^ - Pv^'ut (A.32) h}' - Pv^M^ (A.33) Geometry constraint: = (A t (A.34) Total mass transfer: Nc A, = ^JV< (A.35) i= l In addition to the equations given in the previous section, relationships for pressure drop, liquid holdup, interfacial area, binary mass transfer coefficients and heat transfer coeffi cients are required. These are detailed in Furlonge (2000). ArrENU/XA. PROCESS MODEIS 173 A.3 Reboiler model Lin Vout Fill Figure A.3: Reboiler Assumptions: • instantaneous heat input • liipiid phase reaction • perfect mixing and eipiilibrium Molar balance on component i: i = 1, N c (A.36) (It Liquid and vapour contributions to component holdup: Ml = + M y, i = I,.., Nc (A.37) Enenjy Ixilance: (A.38) (It J = 1 Vapour and liquid contributions to internal energy: Vtot — ~ tray A.39) APPENDIX A. PROCESS MODELS 174 Total volume constraint: Vt.ay = ^ ^ (A.40) Equilibrium relationship: Pi = KiXi 2 = 1, Nc (A.41) Normalisation equations: NC NC = = ^ (A.42) 4 = 1 4 = 1 A.4 Condenser model The condenser model is either operated as a perfect condenser with no subcooling or sub cooling and partial condensing are permitted. When operating as a perfect condenser the equilibrium relationship is appUed. When this assumption is relaxed, the equilibrium relationship is removed and the condenser cooling duty Q c is specified. Vin Lout Figure A.4: Condenser Assumptions: • Negligible material holdup • Negligible pressure drop APPENDIX A. PROCESS MODELS 175 Mass balance Lout = Vin (A.43) = Vi,in i (A.44) Energy balance 0 = — Louth^ — Qc (A.45) Equilibrium relationship (perfect condenser only): yi = KiXi i = 1 , N c (A.46) Normalisation equation (perfect condenser only: NC Y , y i = l (A.47) i = l Pressure: Pin = P (A.48) A.5 Reflux drum model Assumptions: • Variable holdup • Liquid and vapour phase considered • Adiabatic operation • Liquid phase chemical reaction APfENDTXA. PEOCES'^AfODElS' 176 Figure A.5: Reflux Drum Component mole balance: ( C k] Y ur^ :A.49) Liquid and vapour contributions to component holdup: M, = + MtotVi i = 1, --AVa A.50) Energy balance: ^ = knhi - E(-A//«)r, (A.51 Vapour and liquid contributions to internal energy: Utot — — P V vessel (A.52) Total volume constraint: :A.53) E q u i I i b ri u m re I a t i o n s h ip : y, = Kpx, i=l,..,Nc (A.54) APPENDIX A. PROCESS MODELS 177 Normalisation equations: NC NO '^Xi = '^ V i = 1 (A.55) i=l i= l A.6 Accumulator model The model of the accumulator is similar to that of the reflux drum, except that there is no liquid flow leaving the unit, i.e. Lout = 0. A ppendix B Linearisation and scaling methods In this appendix, the procedure used in Chapter 5 to generate the linear models from the non-linear process models is outlined together with the method used to scale the linear models. B .l Linearisation of model equations In this section, the method used by gPROMS (Process Systems Enterprise Ltd., 1999) to generate the linear model in the “LINEARISE” task is outlined. The non-linear process model is expressed as a set of nonlinear differential and algebraic equations of the following form: f(x,x,y,u) =0 (B .l) x{t) and y{t) represent the differential and algebraic variables, respectively, which are determined by the ÿPROMSsimulation. x(t) is the derivative of x{t) with time and u(t) are the input variables, given functions of time. If we consider a point on the solution tradjectory (z*(<), x*(t), y*{t), u*(t)). If we linearise the nonlinear equations at this point we can obtain a linear model of the following from: ^S x + ^êx + ^ ë y + ^ S u = 0 (B.2) OX Ox Oy ou 178 APPENDIX B. LINEARISATION AND SCALING METHODS 179 This can be rearranged to the following: 6x Sx + (B.3) _dx du. 6u _dx dy. Sy Assuming that dx dy is not singular, true for most systems of index 1 or less, then: Sx -1 Sx (B.4) Sy ,dx dy. ,dx du. Su this is the linear form of the original non-linear form of equation (B.l) which is separated into the two state space equations to give: X = Ax + Bu y = Cx + T>u (B.5) The LINEARISE task in gPROMS generates the constant matrices (A, B, C and D) for specified subsets of the input variables(u) and the output variables(?/) B.2 Scaling of the linear models The general, unsealed, statespace representation of the linear process model is: X = A x + B 'u ' (B.6) y' = C x -f C u ' (B.7) where x contains the process states, u' contains the unsealed inputs/ disturbances, y' con tains the unsealed output variables and A, B', C and D' are the unsealed constant matri ces generated, in this work, by the ^PROMS LINEARISEid.sk. As indicated in Chapter 5 APPENDIX B. LINEARISATION AND SCALING METHODS 180 it is necessary to scale the model. When considering the scaling of the model, the vector u[nax contains both the maximum allowable values for the inputs and the expected maxi mum values for the disturbances. Vector y'^ax contains the maximum allowable values for the outputs. The diagonal output scaling matrix Sy with elements syij is defined as: ^ y a — ymax,i ^ ~ I:--fNy (B-8) syij = 0 iJ=l,..,Ny i ^ j (B.9) The diagonal input/ disturbance scaling matrix Su with elements suij is defined as: SUii = '^max,i ^ ~ (B.IO) suij = 0 = (B.ll) The scaled state-space model becomes: X = Ax + Bu (B.12) y — Cx -f D u (B.13) where B = B'5, C = 5 - ’C' D = with u and y being the scaled input and output vectors.